Point-Set Topology

The prefix “locally blah” almost always means that for every \(x\in X\), there exists some neighborhood \(N_x\ni x\) which has property “blah”.

See limit point.

A set \(\mathcal{B}\) is a basis for a topology iff

  • \(\mathcal{B}\) is closed under intersections,
  • Every \(x\in X\) is in some basic set,
  • If \(x\) is in the intersection of two basis sets \(B_1 \cap B_2\), there is a third basic open \(B_3 \ni x\) with \(B_3 \subset B_1 \cap B_2\).

The topology generated by \(\mathcal{B}\) is the following: \(U\subseteq X\) is open iff for each \(x\in U\) there is a basic open \(B\) with \(x\in B \subset U\). Equivalently, every open set is a union of basic open sets.

The boundary of a subset \(A\subseteq X\) is defined as \({{\partial}}A \coloneqq{ \operatorname{cl}}_X(A) \setminus A^\circ\). Equivalently, every point \(p\in A\) intersects both \(A\) and \(X\setminus A\).

Given two topologies \(\tau_1, \tau_2\),

  • \(\tau_1\) is finer/stronger/larger than \(\tau_2\) iff \(\tau_1 \supseteq\tau_2\) (idea: finer resolutions).
  • \(\tau_1\) is coarser/weaker/smaller than \(\tau_2\) iff \(\tau_1 \supseteq\tau_2\).

Two topologies are comparable if either \(\tau_1 \subseteq \tau_2\) or \(\tau_2 \subseteq \tau_1\).

The set of all topologies on a given set \(X\) forms a complete lattice bounded under inclusion:

  • \(\sup(\tau_1, \tau_2) = \tau_1 \cup\tau_2\)
    • The finest topology is the discrete topology \(\tau_{{ 0_{\scriptscriptstyle \uparrow}}} \coloneqq 2^X\), where every set is open.
  • \(\inf(\tau_1, \tau_2) = \left\langle{\tau_1 \cap\tau_2}\right\rangle\), the topology generated by the intersection.
    • The coarsest topology is the indiscrete topology \(\tau_{{ \mathscr \emptyset^{\scriptscriptstyle \downarrow}}} \coloneqq\left\{{\emptyset, X}\right\}\).

If \(f:X\to Y\), then

  • Increasing \(\tau(X)\) or decreasing \(\tau(Y)\) makes it easier for \(f\) to be continuous, i.e. every map continuous with respect to \(\tau_1(X)\) will remain continuous with respect to \(\tau_2(X)\). Writing \(\tau_1(X) \to \tau_2(X) \iff \tau_1(X) \leq \tau_2(X)\),

Link to Diagram

Link to Diagram

  • Decreasing \(\tau(X)\) or increasing \(\tau(Y)\) makes it easier for \(f\) to be an open map.

  • For a fixed \(X\), decreasing \(\tau(X)\) makes it easier for sequences to converge in \(X\).

Write \(\tau_{\mathrm{zar}}(X)\) for the Zariski topology on a space and \(\tau_{{\mathrm{an}}}(X)\) for the classical/Euclidean topology. Then \(\tau_{\mathrm{zar}}({\mathbf{C}}^n) < \tau_{{\mathrm{an}}}({\mathbf{C}}^n)\), i.e. the Zariski topology is strictly weaker than the Euclidean topology and has fewer open sets.

A space \(X\) is disconnected iff

  • There exists a separation of \(X\): a decomposition \(X = U\coprod V\) with \(U, V\) disjoint, open, and nonempty.
    • I.e. \(X\) can not be decomposed as the disjoint union of two proper nonempty sets.
  • The only clopen sets of \(X\) are \(\emptyset, X\) -I.e. \(X\) contains no proper nonempty clopen sets.
  • For \(Y \subseteq X\) a subspace, \(Y\) is disconnected iff \(Y\) is disconnected in the subspace topology. Equivalently, a separation of \(Y\) in \(X\) is a decomposition \(Y = U \coprod V\) with \(U, V\) open in \(Y\) and \begin{align*} { \operatorname{cl}}_{Y}(U) \cap V = \emptyset,\qquad U \cap{ \operatorname{cl}}_{Y}(V) = \emptyset ,\end{align*} so neither set contains a limit point of the other.
  • \(\mathop{\mathrm{Hom}}_{\mathsf{Top}}(X, \left\{{0, 1}\right\}) \cong \left\{{0, 1}\right\}\), i.e. all such continuous functions are constant.

Some examples:

  • \({\mathbf{Q}}\) is disconnected, and \(\pi_0({\mathbf{Q}}) \cong {\mathbf{Q}}\): the only connected components are singletons.

Set \(x\sim y\) iff there exists a connected set \(U\ni x, y\) and take equivalence classes. These classes are the connected components of \(X\).

  • A set \(U \subseteq X\) is closed in \(X\) if and only if its complement \(X\setminus U\) is open.
  • A set \(U\) is closed in \(X\) iff every limit point of \(U\) in \(X\) is contained in \(U\).
  • A set \(U\) in \(X\) is closed in \(X\) iff \({ \operatorname{cl}}_X(U) = U\).
  • If \(Y \subseteq X\) is a subspace and \(U \subseteq Y\), then \(U\) is closed in the subspace \(Y\) iff \(U = Y \cap V\) where \(V\) is closed in \(X\).

A map \(f:X\to Y\) is closed if whenever \(U \subseteq X\) is closed in \(X\), \(f(U) \subseteq Y\) is again closed in \(Y\).

For \(U \subseteq X\), the closure of \(U\) in \(X\) is given by \begin{align*} { \operatorname{cl}}_X(U) = \bigcap_{\substack{ B\supseteq U \\ \text{ closed} }} B ,\end{align*} the intersection of all closed sets in \(X\) containing \(U\). For \(Y\subseteq X\) a subspace containing \(U\), the closure of \(U\) in \(Y\) is \begin{align*} { \operatorname{cl}}_Y(U) = { \operatorname{cl}}_X(U) \cap Y .\end{align*} . 1 In general, we write \(\overline{U} \coloneqq{ \operatorname{cl}}_X(U)\).

An equivalent condition: \(x\in { \operatorname{cl}}_X(U) \iff\) every neighborhood of \(x\) in \(X\) intersects \(U\). 2

A topological space \((X, \tau)\) is compact iff \(X\) is Hausdorff and quasicompact: every open cover has a finite subcover. That is, if \(\left\{{U_{j}}\right\}_{j\in J} \subseteq \tau\) is a collection of open sets such that \(X = \bigcup_{j\in J} U_{j}\), then there exists a finite subset \(J' \subset J\) such that \(X \subseteq \bigcup_{j\in J'} U_{j}\).

A map \(f:X\to Y\) between topological spaces is continuous if and only if whenever \(U \subseteq Y\) is open, \(f ^{-1} (U) \subseteq X\) is open.

A collection of subsets \(\left\{{U_\alpha}\right\}\) of \(X\) is said to cover \(X\) iff \(X = \cup_{\alpha} U_\alpha\). If \(A\subseteq X\) is a subspace, then this collection covers \(A\) iff \(A\subseteq \cup_{\alpha} U_\alpha\).

A subspace \(Q\subset X\) is dense iff every neighborhood of every point in \(x\) intersects \(Q\). Equivalently, \({ \operatorname{cl}}_X(Q) = X\).

A space is first-countable iff every point admits a countable neighborhood basis.

A topological space \(X\) is Hausdorff or \(T_2\) iff points can be separated by disjoint neighborhoods: for every \(p\neq q \in X\) there exist disjoint open sets \(U\ni p\) and \(V\ni q\).

A map \(\iota:A\to B\) is injective if it admits a left inverse \(p:B\to A\) satisfying \(p\circ \iota = \operatorname{id}_A\). Equivalently, \begin{align*} \iota(x) = \iota(y) \in B \implies x = y \in A .\end{align*}

A point \(p\in A\) is interior to \(A\) if there exists a neighborhood \(U\ni p\) that is entirely contained in \(A\).

A point \(p\in A\) is isolated if \(p\) is not a limit point of \(A\). Equivalently, there exists a punctured neighborhood of \(p\) that does not intersect \(A\).

For \(A\subset X\), \(x\) is a limit point of \(A\) if every punctured neighborhood \(P_{x}\) of \(x\) intersects \(A\). I.e., every neighborhood of \(x\) intersects \(A\) at a point other than \(x\). Equivalently, \(x\in { \operatorname{cl}}_{X}(A\setminus\left\{{x}\right\})\).

A space is locally connected iff every neighborhood of every point admits a smaller connected neighborhood. I.e. for all \(x\in X\), for all \(N_x \ni x\), there exists a connected set \(U \subset X\) with \(x\in U\).

A space \(X\) is locally compact iff every \(x\in X\) has a neighborhood contained in a compact subset of \(X\).

A collection of subsets \({\mathcal{S}}\) of \(X\) is locally finite iff each point of \(M\) has a neighborhood that intersects at most finitely many elements of \({\mathcal{S}}\).

A space \(X\) is locally path-connected iff every point in \(X\) admits some path-connected neighborhood. Equivalently, \(X\) admits a basis of path-connected open subsets.

A neighborhood of a point \(x\) is any open set containing \(x\).

A space is normal if any two disjoint closed subsets can be separated by neighborhoods.

If \(p\in X\), a neighborhood basis at \(p\) is a collection \({\mathcal{B}}_{p}\) of neighborhoods of \(p\) such that if \(N_{p}\) is a neighborhood of \(p\), then \(N_{p} \supseteq B\) for at least one \(B\in {\mathcal{B}}_{p}\).

A map \(f:X\to Y\) is an open map (respectively a closed map) if and only if whenever \(U \subseteq X\) is open (resp. closed), \(f(U)\) is again open (resp. closed)>

A refinement of an open cover \({\mathcal{U}}\rightrightarrows X\) is an open cover \({\mathcal{V}}\rightrightarrows X\) such that for every \(V_\beta \in {\mathcal{V}}\), there exists a \(U_\alpha \in {\mathcal{U}}\) such that \(V_\beta \subseteq U_\alpha\) – setting \({\mathcal{V}}\leq {\mathcal{U}}\) iff \({\mathcal{V}}\) refines \({\mathcal{U}}\) yields a preorder on all open covers of \(X\).

A topological space \(X\) is paracompact iff every open cover \({\mathcal{U}}\rightrightarrows X\) admits an locally finite refinement – a division into (potentially more) open subsets \({\mathcal{V}}\rightrightarrows X\) such that each \(x\in X\) is contained in only finitely many \(V_\beta\).

A map \(q:X\to Y\) is a quotient map if and only if

  • \(q\) is surjective, and
  • \(U\) is open in \(Y\iff q ^{-1} (U)\) is open in \(X\)

Note that \(\implies\) comes from the definition of continuity of \(q\), but \(\impliedby\) is a stronger condition.


  • \(p\) maps saturated subsets of \(X\) to open subsets of \(Y\), or
  • If \(U\) is open in \(X\), then \((q^{-1}\circ q)(U)\) is again open in \(X\).

A space \(X\) is path connected if and only if for every pair of points \(x\neq y\) there exists a continuous map \(f:I \to X\) such that \(f(0) = x\) and \(f(1) = y\).

Set \(x\sim y\) iff there exists a path-connected set \(U\ni x, y\), then the equivalence classes are the path components of \(X\).

A subset \(A\subseteq X\) is precompact iff \({ \operatorname{cl}}_{X}(A)\) is compact.

Given a collection of spaces \(\left\{{ (X_i, \tau(X_i) ) }\right\}_{i\in I}\), the box topology is defined by \begin{align*} \tau^{\Box}\qty{ \prod_{i\in I} X_i } &\coloneqq\left\langle{\left\{{\prod_{i\in I} U_i {~\mathrel{\Big\vert}~}U_i \in \tau(X_i) }\right\}}\right\rangle ,\end{align*} the topology generated by products of open sets in the \(X_i\), with no restrictions.

The product topology is defined by \begin{align*} \tau^{\prod}\qty{ \prod_{i\in I} X_i } &\coloneqq\left\{{\prod_{i\in I} U_i {~\mathrel{\Big\vert}~}U_i \in \tau(X_i),\, U_i \neq X_i \text{ for finitely many }i }\right\} ,\end{align*} whose open sets are products of open sets in the \(X_i\) where all but finitely many components are not the entire space \(X_i\). Equivalently, \begin{align*} \tau^\prod\qty{ \prod_{i\in I} X_i} = \inf\left\{{\tau\qty{\prod_{i\in I} X_i } {~\mathrel{\Big\vert}~}p_j: \prod_{i\in I} X_i \to X \text{ are $\tau{\hbox{-}}$continuous for all } j}\right\} ,\end{align*} the smallest/coarsest/initial topology such that the projections \(p_j: \prod_i X_i \to X_j\) are continuous. Equivalently, \begin{align*} \tau^\prod\qty{\prod_{i\in I} X_i} = \left\langle{\left\{{ p_i^{-1}(U_i) {~\mathrel{\Big\vert}~}U_i\in \tau(X_i)}\right\}_{i\in I} }\right\rangle ,\end{align*} the topology generated by basic open sets which are preimages of opens in the components under the canonical projections.

Equivalently, it is the terminal cone over all of the \(X_i\) in \({\mathsf{Top}}\):

Link to Diagram

Note that \begin{align*} \tau^{\Box}(X) \geq \tau^{\prod}(X) ,\end{align*} i.e. the box topology is finer and has more open sets, making convergence harder in the box topology. The product topology is preferred since continuous maps \(f: Y\to \prod X_i\) into the product can be given by continuous component maps \(f_i: Y\to X_i\).

Several equivalent definitions. Let \(f: X\to Y\) be continuous, then \(f\) is proper iff

  • Most general: preimages of compact sets are compact: if \(K \subseteq Y\) is compact, then \(f^{-1}(K) \subseteq X\) is compact.

  • For \(Y\) Hausdorff and locally compact, \(f\) is a closed map with compact fibers: \(f^{-1}(\left\{{y}\right\})\) is compact for every \(y\in Y\).

  • For \(X\) Hausdorff and \(Y\) locally compact, \(f\) is universally closed: the map \(f\times \operatorname{id}_Z: X\times Z\to Y\times Z\) is a closed map for every space \(Z\).

  • For \(X, Y\) metric spaces, if \(\left\{{x_i}\right\}\) is a sequence that eventually escapes every compact set in \(X\), \(\left\{{f(x_i)}\right\}\) eventually escapes every compact set in \(Y\).

A topological space \(X\) (possible non-Hausdorff) is quasicompact iff every open cover admits a finite subcover. If \(X\) is additionally Hausdorff, \(X\) is said to be compact.

A cover \({\mathcal{V}}\rightrightarrows X\) is a refinement of \({\mathcal{U}}\rightrightarrows X\) iff for each \(V\in {\mathcal{V}}\) there exists a \(U\in{\mathcal{U}}\) such that \(V\subseteq U\).

A space \(X\) is regular if whenever \(x\in X\) and \(F\not\ni x\) is closed, \(F\) and \(x\) are separated by neighborhoods.

A retract \(r\) of \(B\) onto a subspace \(A\) is a map \(r:B\to A\) that is a left-inverse for the inclusion \(f:A\hookrightarrow B\), so \(r \circ f = \operatorname{id}_A\):

Link to (partial) Diagram

Equivalently, a continuous map \(r:B\to A\) with \({ \left.{{r}} \right|_{{A}} } = \operatorname{id}_A\) restricting to the identity on \(A\), i.e. fixing \(A\) pointwise. Note that \(r\) is necessarily a surjection.

Alt: Let \(X\) be a topological space and \(A \subset X\) be a subspace, then a retraction of \(X\) onto \(A\) is a map \(r: X\to X\) such that the image of \(X\) is \(A\) and \(r\) restricted to \(A\) is the identity.

If \(X\) retracts onto \(A\) with \(\iota:A\hookrightarrow X\), then \(i_*\) is injective. Any nonempty space retracts to a point via a constant map.

A subset \(U \subseteq X\) is saturated with respect to a surjective map \(p: X\twoheadrightarrow Y\) if and only if whenever \(U \cap p ^{-1} (y) = V \neq \emptyset\), we have \(V \subseteq U\), i.e. \(U\) contains every set \(p ^{-1} (y)\) that it intersects. Equivalently, \(U\) is the complete inverse image of a subset of \(Y\).

A space \(X\) is separable iff \(X\) contains a countable dense subset.

A space is second-countable iff it admits a countable basis.

For \((X, \tau)\) a topological space and \(U \subseteq X\) an arbitrary subset, the space \((U, \tau_U)\) is a topological space with a subspace topology defined by \begin{align*} \tau_U \coloneqq\left\{{Y \cap U {~\mathrel{\Big\vert}~}U \in \tau}\right\} .\end{align*}

A map \(\pi\) with a right inverse \(f\) satisfying \begin{align*}\pi \circ f = \operatorname{id}\end{align*}

  • \(T_0\): points are topologically distinguishable, i.e. for any 2 points \(x_1\neq x_2\), at least one \(x_i\) (say \(x_1\)) admits a neighborhood not containing \(x_2\).

  • \(T_1\): For any 2 points, both admit neighborhoods not containing the other. Equivalently, points are closed.

  • \(T_2\): For any 2 points, both admit disjoint separating neighborhoods.

  • \(T_{2.5}\): For any 2 points, both admit disjoint closed separating neighborhoods.

  • \(T_3\): \(T_0\) & regular. Given any point \(x\) and any closed \(F\not\ni x\), there are neighborhoods separating \(F\) and \(x\).

  • \(T_{3.5}\): \(T_0\) & completely regular. Any point \(x\) and closed \(F\not\ni x\) can be separated by a continuous function.

  • \(T_4\): \(T_1\) & normal. Any two disjoint closed subsets can be separated by neighborhoods.

  • Not \(T_0\): the space \(\left\{{ f:{\mathbf{R}}\to {\mathbf{C}}{~\mathrel{\Big\vert}~}\int_{\mathbf{R}}{\left\lvert {f} \right\rvert}^2 < \infty }\right\}\), since two a.e. equal functions aren’t distinguishable (they have precisely the same set of neighborhoods).
  • \(T_1\) but not \(T_0\): \(\operatorname{Spec}R\) for \(R\in \mathsf{CRing}\) with the Zariski topology. There are points that aren’t closed: \(\operatorname{Spec}R \setminus\operatorname{mSpec}R\).

  • Using open sets: closed under arbitrary unions and finite intersections.
  • Using closed sets: closed under arbitrary intersections and finite unions.

A mnemonic: in \({\mathbf{R}}\), \(\cap_{n\in {\mathbb{N}}} (-1/n, 1/n) = \left\{{0}\right\}\) which is closed in \({\mathbf{R}}\).

A topological embedding is a continuous map \(f:X\to Y\) which is a homeomorphism onto its image, i.e. \(X\cong_{{\mathsf{Top}}} f(X)\).

Analysis and Metric Spaces

For a subset \(A\) of a metric space \((X, d)\), the diameter of \(A\) is defined as \begin{align*} {\operatorname{diam}}(A) \coloneqq\sup_{p, q\in A}d(p, q) .\end{align*}

For \((X, d)\) a metric space, \(S \subset X\), and \(f\in X\), the distance from \(f\) to \(S\) is \begin{align*} \operatorname{dist}(f, S) \coloneqq\inf_{s\in S} d(f, s) .\end{align*}

A set \(S\) in a metric space \((X, d)\) is bounded iff there exists an \(m\in {\mathbf{R}}\) such that \(d(x, y) < m\) for every \(x, y\in S\).

For \(f: (X, d_{x}) \to (Y, d_{Y})\) metric spaces, \(f\) is uniformly continuous iff \begin{align*} \forall {\varepsilon}> 0, ~\exists \delta > 0 \text{ such that } \quad d_{X}(x_{1}, x_{2}) < \delta \implies d_{Y}(f(x_{1}), f(x_{2})) < {\varepsilon} .\end{align*}

For \((X, d)\) a compact metric space and \(\left\{{U_\alpha}\right\}\rightrightarrows X\), there exists a Lebesgue number \(\delta_{L} > 0\) which satisfies \begin{align*} A\subset X, ~ {\operatorname{diam}}(A) < \delta_{L} \implies A\subseteq U_\alpha \text{ for some } \alpha .\end{align*}

Algebraic Topology


Points \(x\in M^n\) defined by \begin{align*} {\partial}M = \left\{{x\in M: H_{n}(M, M-\left\{{x}\right\}; {\mathbf{Z}}) = 0}\right\} \end{align*}

Denoting \(\Delta^p \xrightarrow{\sigma} X \in C_{p}(X; G)\), a map that sends pairs (\(p{\hbox{-}}\)chains, \(q{\hbox{-}}\)cochains) to \((p-q){\hbox{-}}\)chains \(\Delta^{p-q} \to X\) by \begin{align*} H_{p}(X; R)\times H^q(X; R) \xrightarrow{\frown} H_{p-q}(X; R)\\ \sigma \frown \psi = \psi(F_{0}^q(\sigma))F_{q}^p(\sigma) \end{align*} where \(F_{i}^j\) is the face operator, which acts on a simplicial map \(\sigma\) by restriction to the face spanned by \([v_{i} \ldots v_{j}]\), i.e. \(F_{i}^j(\sigma) = {\left.{{\sigma}} \right|_{{[v_{i} \ldots v_{j}]}} }\).


A map \(X \xrightarrow{f} Y\) is said to be cellular if \(f(X^{(n)}) \subseteq Y^{(n)}\) where \(X^{(n)}\) denotes the \(n{\hbox{-}}\) skeleton.

A manifold that is compact, with or without boundary.

A constant map \(f: X\to Y\) iff \(f(X) = y_{0}\) for some \(y_{0}\in Y\), i.e. for every \(x\in X\) the output value \(f(x) = y_{0}\) is the same.

For a directed system \((X_{i}, f_{ij})\), the colimit is an object \(X\) with a sequence of projections \(\pi_{i}:X\to X_{i}\) such that for any \(Y\) mapping into the system, the following diagram commutes:

  • Products
  • Pullbacks
  • Inverse / projective limits
  • The \(p{\hbox{-}}\)adic integers \({\mathbf{Z}}_{p}\).

A space \(X\) is contractible if \(\operatorname{id}_X\) is nullhomotopic. i.e. the identity is homotopic to a constant map \(c(x) = x_0\).

Equivalently, \(X\) is contractible if \(X \simeq\left\{{x_0}\right\}\) is homotopy equivalent to a point. This means that there exists a mutually inverse pair of maps \(f: X \to\left\{{x_0}\right\}\) and \(g:\left\{{x_0}\right\} \to X\) such that \(f\circ g \simeq\operatorname{id}_{\left\{{x_0}\right\}}\) and \(g\circ f \simeq\operatorname{id}_X\). 3


A covering space of \(X\) is the data \(p: \tilde X \to X\) such that

  • Each \(x\in X\) admits a neighborhood \(U\) such that \(p ^{-1} (U)\) is a union of disjoint open sets in \(\tilde V_i \subseteq X\) (the sheets of \(\tilde X\) over \(U\)),
  • \({ \left.{{p}} \right|_{{V_i}} }: V_i \to U\) is a homeomorphism for each sheet.

An isomorphism of covering spaces \(\tilde X_1 \cong \tilde X_2\) is a commutative diagram

Link to diagram

A map taking pairs (\(p{\hbox{-}}\)cocycles, \(q{\hbox{-}}\)cocycles) to \((p+q){\hbox{-}}\)cocyles by \begin{align*} H^p(X; R) \times H^q(X; R) \xrightarrow{\smile} H^{p+q}(X; R)\\ (a \cup b)(\sigma) = a(\sigma \circ I_{0}^p)~b(\sigma \circ I_{p}^{p+q}) \end{align*} where \(\Delta^{p+q} \xrightarrow{\sigma} X\) is a singular \(p+q\) simplex and

\begin{align*}I_{i}^j: [i, \cdots, j] \hookrightarrow\Delta^{p+q} .\end{align*}

is an embedding of the \((j-i){\hbox{-}}\)simplex into a \((p+q){\hbox{-}}\)simplex.

On a manifold, the cup product is Poincaré dual to the intersection of submanifolds. Also used to show \(T^2 \not\simeq S^2 \vee S^1 \vee S^1\).


An \(n{\hbox{-}}\)cell of \(X\), say \(e^n\), is the image of a map \(\Phi: B^n \to X\). That is, \(e^n = \Phi(B^n)\). Attaching an \(n{\hbox{-}}\)cell to \(X\) is equivalent to forming the space \(B^n \coprod_{f} X\) where \(f: {\partial}B^n \to X\).

  • A \(0{\hbox{-}}\)cell is a point.
  • A \(1{\hbox{-}}\)cell is an interval \([-1, 1] = B^1 \subset {\mathbf{R}}^1\). Attaching requires a map from \(S^0 =\left\{{-1, +1}\right\} \to X\)
  • A \(2{\hbox{-}}\)cell is a solid disk \(B^2 \subset {\mathbf{R}}^2\) in the plane. Attaching requires a map \(S^1 \to X\).
  • A \(3{\hbox{-}}\)cell is a solid ball \(B^3 \subset {\mathbf{R}}^3\). Attaching requires a map from the sphere \(S^2 \to X\).

Letting \(\mathsf{C} \coloneqq\mathsf{Cov}(X) \leq {\mathsf{Top}}_{/ {X}}\) be the subcategory of the slice category over \(X\) of covering maps \(\tilde X\to X\), the group of deck transformations is given by \begin{align*} \mathrm{Deck}(\tilde X\to X) \coloneqq\mathop{\mathrm{Aut}}_{\mathsf{C}}(\tilde X\to X) ,\end{align*} i.e topological automorphisms of \(\tilde X\) which fix \(X\) pointwise.


A map \(r\) in \(A\mathrel{\textstyle\substack{\hookrightarrow^{\iota}\\\textstyle\dashleftarrow_{r}}} X\) that is a retraction (so \(r\circ \iota = \operatorname{id}_{A}\)) that also satisfies \(\iota \circ r \simeq\operatorname{id}_{X}\).

Note that this is equality in one direction, but only homotopy equivalence in the other.

Equivalently, a map \(F:I\times X\to X\) such that \begin{align*} F_{0}(x) &= \operatorname{id}_{X} F_{t}(x)\mathrel{\Big|}_{A} &= \operatorname{id}_{A} F_{1}(X) &= A .\end{align*}


A deformation retract is a homotopy \(H:X\times I \to X\) from \(\operatorname{id}_X\) to \(\operatorname{id}_A\) where \({ \left.{{H}} \right|_{{A}} } = \operatorname{id}_A\) fixes \(A\) at all times. \begin{align*} H: X\times I \to X \\ H(x, 0) = \operatorname{id}_X \\ H(x, 1) = \operatorname{id}_A \\ x\in A \implies H(x, t) \in A \quad \forall t \end{align*}

A deformation retract between a space and a subspace is a homotopy equivalence, and further \(X\simeq Y\) iff there is a \(Z\) such that both \(X\) and \(Y\) are deformation retracts of \(Z\). Moreover, if \(A\) and \(B\) both have deformation retracts onto a common space \(X\), then \(A \simeq B\).

Given any \(f: S^n \to S^n\), there are induced maps on homotopy and homology groups. Taking \(f^*: H^n(S^n) \to H^n(S^n)\) and identifying \(H^n(S^n) \cong {\mathbf{Z}}\), we have \(f^*: {\mathbf{Z}}\to{\mathbf{Z}}\). But homomorphisms of free groups are entirely determined by their action on generators. So if \(f^*(1) = n\), define \(n\) to be the degree of \(f\), which only depends on the homotopy class \(f\in [S^n, S^n]\).

For \(x\in M\), the only nonvanishing homology group \(H_{i}(M, M - \left\{{x}\right\}; {\mathbf{Z}})\)


An action \(G\curvearrowright X\) is properly discontinuous if each \(x\in X\) has a neighborhood \(U\) such that all of the images \(g(U)\) for \(g\in G\) are disjoint, i.e. \(g_1(U) \cap g_2(U) \neq \emptyset \implies g_1 = g_2\). The action is free if there are no fixed points.

Sometimes a slightly weaker condition is used: every point \(x\in X\) has a neighborhood \(U\) such that \(U \cap G(U) \neq \emptyset\) for only finitely many \(G\).


For a connected, closed, orientable manifold, \([M]\) is a generator of \(H_{n}(M; {\mathbf{Z}}) = {\mathbf{Z}}\).


Let \(X, Y\) be topological spaces and \(f,g: X \to Y\) continuous maps. Then a homotopy from \(f\) to \(g\) is a continuous function

\(F: X \times I \to Y\)

such that

\(F(x, 0) = f(x)\) and \(F(x,1) = g(x)\)

for all \(x\in X\). If such a homotopy exists, we write \(f\simeq g\). This is an equivalence relation on \(\text{Hom}(X,Y)\), and the set of such classes is denoted \([X,Y] \coloneqq\hom (X,Y)/\simeq\).


Let \(f: X \to Y\) be a continuous map, then \(f\) is said to be a homotopy equivalence if there exists a continuous map \(g: X \to Y\) such that

\(f\circ g \simeq\operatorname{id}_Y\) and \(g\circ f \simeq\operatorname{id}_X\).

Such a map \(g\) is called a homotopy inverse of \(f\), the pair of maps is a homotopy equivalence.

If such an \(f\) exists, we write \(X \simeq Y\) and say \(X\) and \(Y\) have the same homotopy type, or that they are homotopy equivalent.

Note that homotopy equivalence is strictly weaker than homeomorphic equivalence, i.e., \(X\cong Y\) implies \(X \simeq Y\) but not necessarily the converse.


For a manifold \(M\), a map on homology defined by \begin{align*} H_{\widehat{i}}M \otimes H_{\widehat{j}}M \to H_{\widehat{i+j}}X\\ \alpha\otimes\beta \mapsto \left< \alpha, \beta \right> \end{align*} obtained by conjugating the cup product with Poincaré Duality, i.e. 

\begin{align*}\left< \alpha, \beta \right> = [M] \frown ([\alpha] {}^{ \vee }\smile [\beta] {}^{ \vee }) .\end{align*}

Then, if \([A], [B]\) are transversely intersecting submanifolds representing \(\alpha, \beta\), then \begin{align*}\left<\alpha, \beta\right> = [A\cap B]\end{align*} . If \(\widehat{i} = j\) then \(\left< \alpha, \beta \right> \in H_{0} M = {\mathbf{Z}}\) is the signed number of intersection points.

Alt: The pairing obtained from dualizing Poincare Duality to obtain \begin{align*}\mathrm{F}(H_{i} M) \otimes\mathrm{F}(H_{n-i}M) \to {\mathbf{Z}}\end{align*} Computed as an oriented intersection number between two homology classes (perturbed to be transverse).

The nondegenerate bilinear form cohomology induced by the Kronecker Pairing: \begin{align*}I: H^k(M_{n}) \times H^{n-k}(M^n) \to {\mathbf{Z}}\end{align*} where \(n=2k\).

  • When \(k\) is odd, \(I\) is skew-symmetric and thus a symplectic form.
  • When \(k\) is even (and thus \(n \equiv 0 \operatorname{mod}4\)) this is a symmetric form.
  • Satisfies \(I(x,y) = (-1)^{k(n-k)} I(y, x)\)

A map pairing a chain with a cochain, given by \begin{align*} H^n(X; R) \times H_{n}(X; R) \to R \\ ([\psi, \alpha]) \mapsto \psi(\alpha) \end{align*} which is a nondegenerate bilinear form.


At a point \(x \in V \subset M\), a generator of \(H_{n}(V, V-\left\{{x}\right\})\). The degree of a map \(S^n \to S^n\) is the sum of its local degrees.


\(H_{n}(X, X-A; {\mathbf{Z}})\) is the local homology at \(A\), also denoted \(H_{n}(X \mathrel{\Big|}A)\)

At a point \(x\in M^n\), a choice of a generator \(\mu_{x}\) of \(H_{n}(M, M - \left\{{x}\right\}) = {\mathbf{Z}}\).

An \(n{\hbox{-}}\)manifold is a Hausdorff space in which each neighborhood has an open neighborhood homeomorphic to \({\mathbf{R}}^n\).

A manifold in which open neighborhoods may be isomorphic to either \({\mathbf{R}}^n\) or a half-space \(\left\{{\mathbf{x} \in {\mathbf{R}}^n \mathrel{\Big|}x_{i} > 0}\right\}\).


A covering space is normal if and only if for every \(x\in X\) and every pair of lifts \(\tilde x_1, \tilde x_2\), there is a deck transformation \(f\) such that \(f(\tilde x_1) = \tilde x_2\).

A map \(X\xrightarrow{f} Y\) is nullhomotopic if it is homotopic to a constant map \(X \xrightarrow{g} \left\{{y_{0}}\right\}\); that is, there exists a homotopy \begin{align*} F: X\times I &\to Y \\ {\left.{{F}} \right|_{{X\times\left\{{0}\right\}}} } &= f \quad F(x, 0) = f(x) \\ {\left.{{F}} \right|_{{X\times\left\{{1}\right\}}} } &= g \quad F(x, 1) = g(x) = y_{0}\\ .\end{align*}


If \(f\) is homotopic to a constant map, say \(f: x \mapsto y_0\) for some fixed \(y_0 \in Y\), then \(f\) is said to be nullhomotopic. In other words, if \(f:X\to Y\) is nullhomotopic, then there exists a homotopy \(H: X\times I \to Y\) such that \(H(x, 0) = f(x)\) and \(H(x, 1) = y_0\).

Note that constant maps (or anything homotopic) induce zero homomorphisms.

For a group action \(G\curvearrowright X\), the orbit space \(X/G\) is defined as \(X/\sim\) where \(\forall x,y\in X, x\sim y \iff \exists g\in G \mathrel{\Big|}g.x = y\).

A manifold for which an orientation exists, see “Orientation of a Manifold”.

For any manifold \(M\), a two sheeted orientable covering space \(\tilde M_{o}\). \(M\) is orientable iff \(\tilde M\) is disconnected. Constructed as \begin{align*} \tilde M = \coprod_{x\in M}\left\{{\mu_{x} \mathrel{\Big|}\mu_{x}~ \text{is a local orientation}}\right\} .\end{align*}

A family of \(\left\{{\mu_{x}}\right\}_{x\in M}\) with local consistency: if \(x,y \in U\) then \(\mu_{x}, \mu_{y}\) are related via a propagation.

Formally, a function \begin{align*}M^n \to \coprod_{x\in M} H(X \mathrel{\Big|}\left\{{x}\right\})\\ x \mapsto \mu_{x}\end{align*} such that \(\forall x \exists N_{x}\) in which \(\forall y\in N_{x}\), the preimage of each \(\mu_{y}\) under the map \(H_{n}(M\mathrel{\Big|}N_{x}) \twoheadrightarrow H_{n}(M\mathrel{\Big|}y)\) is a single generator \(\mu_{N_{x}}\).


  • \(M\) is orientable.
  • The map \(W: (M, x) \to {\mathbf{Z}}_{2}\) is trivial.
  • \(\tilde M_{o} = M \coprod {\mathbf{Z}}_{2}\) (two sheets).
  • \(\tilde M_{o}\) is disconnected
  • The projection \(\tilde M_{o} \to M\) admits a section.

For a closed, orientable \(n{\hbox{-}}\)manifold, following map \([M] \frown {-}\) is an isomorphism: \begin{align*} D: H^k(M; R) \to H_{n-k}(M; R) \\ D(\alpha) = [M] \frown \alpha\end{align*}


A space \(X\) is semilocally simply connected if every \(x\in X\) has a neighborhood \(U\) such that \(U\hookrightarrow X\) induces the trivial map \(\pi_1(U;x) \to \pi_1(X, x)\).


Given a simplex \(\sigma = [v_1 \cdots v_n]\), define the face map \begin{align*} {\partial}_i:\Delta^n &\to \Delta^{n-1} \\ \sigma &\mapsto [v_1 \cdots \widehat{v}_i \cdots v_n] \end{align*}

A simplicial complex is a set \(K\) satisfying

  • \(\sigma \in K \implies {\partial}_i\sigma \in K\).

  • \(\sigma,\tau\in K \implies \sigma\cap\tau = \emptyset,~ {\partial}_i\sigma,~\text{or}~{\partial}_i\tau\).

This amounts to saying that any collection of \((n-1)\)-simplices uniquely determines an \(n\)-simplex (or its lack thereof), or that that map \(\Delta^k \to X\) is a continuous injection from the standard simplex in \({\mathbf{R}}^n\).

  • \({\left\lvert {K\cap B_\varepsilon(\sigma)} \right\rvert} < \infty\) for every \(\sigma\in K\), identifying \(\sigma \subseteq {\mathbf{R}}^n\).

For a map \begin{align*}K\xrightarrow{f} L\end{align*} between simplicial complexes, \(f\) is a simplicial map if for any set of vertices \(\left\{{v_{i}}\right\}\) spanning a simplex in \(K\), the set \(\left\{{f(v_{i})}\right\}\) are the vertices of a simplex in \(L\).

A space \(X\) is simply connected if and only if \(X\) is path-connected and every loop \(\gamma : S^1 \xrightarrow{} X\) can be contracted to a point.

Equivalently, there exists a lift \(\widehat{\gamma }: D^2 \xrightarrow{} X\) such that \({ \left.{{\widehat{\gamma}}} \right|_{{{{\partial}}D^2}} } = \gamma\).

Equivalently, for any two paths \(p_1, p_2: I \xrightarrow{} X\) such that \(p_1(0) = p_2(0)\) and \(p_1(1) = p_2(1)\), there exists a homotopy \(F: I^2 \xrightarrow{} X\) such that \({ \left.{{F}} \right|_{{0}} } = p_1,\, { \left.{{F}} \right|_{{0}} } = p_2\).

Equivalently, \(\pi _1 X = 1\) is trivial.

\begin{align*}x \in C_{n}(x) \implies X = \sum_{i} n_{i} \sigma_{i} = \sum_{i} n_{i} (\Delta^n \xrightarrow{\sigma_{i}} X) .\end{align*}

\begin{align*}x \in C^n(x) \implies X = \sum_{i} n_{i} \psi_{i} = \sum_{i} n_{i} (\sigma_{i} \xrightarrow{\psi_{i}} X) .\end{align*}



For a space \(X\), defined as \begin{align*} CX = \frac{X\times I} {X \times\left\{{0}\right\}} .\end{align*} Example: The cone on the circle \(CS^1\)

Note that the cone embeds \(X\) in a contractible space \(CX\).

Compact represented as \(\Sigma X = CX \coprod_{\operatorname{id}_{X}} CX\), two cones on \(X\) glued along \(X\). Explicitly given by

\begin{align*}\Sigma X = \frac{X\times I}{(X\times\left\{{0}\right\}) \cup(X\times\left\{{1}\right\}) \cup(\left\{{x_{0}}\right\} \times I)} .\end{align*}


Homological Algebra

For an \(R{\hbox{-}}\)module \begin{align*} \operatorname{Tor}_{R}^n({-}, B) = L_{n}({-}\otimes_{R} B) ,\end{align*} where \(L_{n}\) denotes the \(n\)th left derived functor.


\(S = \left\{{s_{i}}\right\}\) is a generating set for an \(R{\hbox{-}}\) module \(M\) iff \begin{align*}x\in M \implies x = \sum r_{i} s_{i}\end{align*} for some coefficients \(r_{i} \in R\) (where this sum may be infinite).

For an \(R{\hbox{-}}\)module \(M\), a basis \(B\) is a linearly independent generating set. An \(R{\hbox{-}}\)module is free iff it admits a basis.

An element \(c \in C_{p}(X; R)\) can be represented as the singular \(p\) simplex \(\Delta^p \to X\).

Given two maps between chain complexes \((C_*, {\partial}_{C}) \xrightarrow{f, ~g} (D_*, {\partial}_{D})\), a chain homotopy is a family \(h_{i}: C_{i} \to B_{i+1}\) satisfying \begin{align*}f_{i}-g_{i} = {\partial}_{B, i-1}\circ h_{n} + h_{i+1}\circ {\partial}_{A, i}\end{align*} .

A map between chain complexes \((C_*, {\partial}_{C}) \xrightarrow{f} (D_*, {\partial}_{D})\) is a chain map iff each component \(C_{i} \xrightarrow{f_{i}} D_{i}\) satisfies \begin{align*} f_{i-1}\circ{\partial}_{C, i} = {\partial}_{D,i} \circ f_{i} \end{align*} (i.e this forms a commuting ladder)


An cochain \(c \in C^p(X; R)\) is a map \(c \in \hom(C_{p}(X; R), R)\) on chains.


For a functor \(T\) and an \(R{\hbox{-}}\)module \(A\), a left derived functor \((L_{nT})\) is defined as \(h_{n}(TP_{A})\), where \(P_{A}\) is a projective resolution of \(A\).


A functor \(T\) is right exact if a short exact sequence

\begin{align*}0 \to A \to B \to C \to 0 \end{align*} yields an exact sequence

\begin{align*}\ldots TA \to TB \to TC \to 0 \end{align*} and is left exact if it yields

\begin{align*}0 \to TA \to TB \to TC \to \ldots \end{align*} Thus a functor is exact iff it is both left and right exact, yielding

\begin{align*}0 \to TA \to TB \to TC \to 0 .\end{align*}

\({-}\otimes_{R} {-}\) is a right exact bifunctor.


An \(R{\hbox{-}}\)module is flat if \(A\otimes_{R} {-}\) is an exact functor.

A \({\hbox{-}}\)module \(M\) with a basis \(S = \left\{{s_{i}}\right\}\) of generating elements. Every such module is the image of a unique map \(\mathcal{F}(S) = R^S \twoheadrightarrow M\), and if \(M = \left< S \mathrel{\Big|}\mathcal{R} \right>\) for some set of relations \(\mathcal{R}\), then \(M \cong R^S / \mathcal{R}\).


A generating \(S\) for a module \(M\) is linearly independent if \(\sum r_{i} s_{i} = 0_M \implies \forall i,~r_{i} = 0\) where \(s_{i}\in S, r_{i} \in R\).


A pairing alone is an \(R{\hbox{-}}\)bilinear module map, or equivalently a map out of a tensor product since \(p: M\otimes_{R} N \to L\) can be partially applied to yield \(\phi: M \to L^N = \hom_{R}(N, L)\). A pairing is perfect when \(\phi\) is an isomorphism.

This is theorem 17.4 in Munkres
Munkres 17.5
This is a useful property because it supplies you with a homotopy.