Tags: #qual_topology
Topics

Definitions:
 topologies,
 open/closed/clopen, bases,
 continuity,
 homeomorphisms,
 subspaces
 products,
 quotients
 closures,
 retracts

Metric spaces
 Complete
 Bounded
 Compactness

Connectedness
 Pathconnected
 Locally pathconnected
 Totally disconnected

Separation axioms,
 Hausdorff,
 Normal,
 Regular
 The tube lemma
 Common counterexamples (sine curve)
Warmups
 State the axioms of a topology.
 What does it mean for a set to be open? Closed?
 State the definition of the product topology, the subspace topology, and the quotient topology.
 What does it mean for a family of sets to form a basis for a topology?
 What is an interior point? An isolated point? A limit point?
 What is the closure of a subspace \(E\subseteq X\)?
 What does it mean for a topological space to be compact?
 What does it mean for \(E\subseteq X\) to be a dense subspace?
 Come up with 6 different topologies on \({\mathbf{R}}^d\).
 What is a separable space?
 What is a nowhere dense subspace?
Exercises
 Prove Cantor’s intersection theorem.

Determine if the following subsets of \({\mathbf{R}}\) are opened, closed, both, or neither:
 \({\mathbf{Q}}\)
 \({\mathbf{Z}}\)
 \(\left\{{1}\right\}\)
 \(\left\{{p \in {\mathbf{Z}}^{\geq 0} {~\mathrel{\Big\vert}~}p\text{ is prime}}\right\}\)
 \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\}\)
 \(\left\{{ {1\over n} {~\mathrel{\Big\vert}~}n\in {\mathbf{Z}}^{\geq 0}}\right\} \cup\left\{{0}\right\}\)
 Prove that \({\mathbf{R}}^n\) is not homeomorphic to \({\mathbf{R}}\) for any \(n\geq 2\).

Is it true that the closure of a product is the product of the closures?
 Is it true that the interior of a product is the product of the interiors?
 Find a space that is connected but not locally connected. Can there be a space that is locally connected but not connected?
 Show that for \(X\) an arbitrary topological space, the onepoint compactification \(\widehat{X}\) (with its corresponding topology) is compact.

Prove that pathconnected implies connected
 Show that the topologist’s sine curve is connected but not pathconnected.
 Is every product (finite or infinite) of Hausdorff spaces Hausdorff?
 Is \({\mathbf{R}}\) homeomorphic to \([0, \infty)\)?
 Show that \(X\) is connected iff the only subsets of \(X\) which are both closed and open are \(\emptyset, X\).
 Show that a closed subset \(A\) of a compact space \(X\) is compact. Does this hold when \(A\) is instead an open subset?
 Show that if \(f:X\to Y\) is continuous and \(X\) is compact then the image \(f(X)\subseteq Y\) is compact.
 Show that every compact metric space is complete.

Show that a compact subset of a Hausdorff space is closed. Does the converse hold?
 What property on a space guarantees that compact sets are closed
 What property on a space guarantees that closed sets are compact?
 Show that a continuous bijection from a compact space to a Hausdorff space is necessarily a homeomorphism.
 Prove the following implications of separation axioms, and show that they are strict:
 Show that every compact metrizable space has a countable basis.
Qual Questions
Tube lemma: