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 \({\mathbb{R}}^d\).
What is a separable space?
What is a nowhere dense subspace?

Exercises

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