# 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.

Tube lemma: