Topology Qual Prep Week 1: Point-Set

Tags: #qual_topology #qualifying_exam


  • Definitions:
    • topologies,
    • open/closed/clopen, bases,
    • continuity,
    • homeomorphisms,
    • subspaces
    • products,
    • quotients
    • closures,
    • retracts
  • Metric spaces
    • Complete
    • Bounded
  • Compactness
  • Connectedness
    • Path-connected
    • Locally path-connected
    • Totally disconnected
  • Separation axioms,
    • Hausdorff,
    • Normal,
    • Regular
  • The tube lemma
  • Common counterexamples (sine curve)


  • 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?


  • 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 one-point compactification \(\widehat{X}\) (with its corresponding topology) is compact.
  • Prove that path-connected implies connected
    • Show that the topologist’s sine curve is connected but not path-connected.
  • 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:

#qual_topology #qualifying_exam