Tags: #qual_topology #qualifying_exam

Topics

• 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)

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 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: