## Math 4121a/9021a -- General Topology

Instructor Graham Denham To be announced Tuesday, Thursday 1:00-2:20pm MC 108 General Topology, Stephen Willard (Dover edition). Mathematics 3122A/B or permission of instructor. Solutions December 10th, 2-5pm, MC 108; alternate December 16th, 7-10pm, SH 3307 40% Final exam; 30% midterm; 30% homework

## Important

The first day of class will be September 9th!  The time and location will be determined at the Math scheduling meeting: 10am, MC 107, Thursday September 4th.  It may be possible to take the course without the usual Metric Space Topology prerequisite, depending on your circumstances.

## Synopsis

What's happening?
• 9/9: introduction and somewhat informal Zermelo-Fraenkel set theory.
• 9/11: metric spaces.  We define open sets in metric spaces to generalize the notion from Euclidean space.  We see that the "$$\epsilon,\delta$$" definition of continuity can be replaced by an equivalent statement involving images of open sets.
• 9/16: topological spaces in the abstract.  Examples: topology induced by a metric; discrete, indiscrete, cofinite topologies... closure and interior. We see examples where a set admits more than one topology and use the words "weaker" and "finer" if we can compare them.
• 9/18: neighbourhood bases capture the notion of "local"
• 9/23: The idea of a basis for the topology is the global version and provides a convenient way to specify a topology. Examples: the profinite topology, the Zariski topology, the Sorgenfrey Line, and various order topologies.
• 9/25: (cancelled due to illness)
• 9/30: continuity and homeomorphisms.  Not all continuous bijections have continuous inverses!
• 10/2: the product topology seems to be better-behaved than the naive (box) topology on an infinite product.  Signature property: it is the weakest topology that makes coordinate projections continuous.
• 10/7: quotient maps and the quotient (strong) topology.
• 10/9: identification spaces, covering spaces versus local homeomorphisms
• 10/14: sequences: we defined convergence in general and noted that continuous functions preserve limits.  The limit point of a convergent sequence in a set $$E$$ lies in $$\overline{E}$$.  However, $${\mathbf R}^{\mathbf R}$$ with the product topology shows that sequences are not enough, in general, to recover the topology.
• 10/16: sequences work fine for first-countable spaces.  More generally, we use nets.
• 10/21: separability axioms: $$T_0$$ and $$T_1$$.
• 10/23: separability axioms: $$T_2$$ (Hausdorff) -- this can be characterized in terms of uniqueness of limits as well as the image of the diagonal map $$X\to X\times X$$.  Not preserved by arbitrary quotients.  Regular and $$T_3$$.
• 10/28: separability axioms: $$T_4$$ is "normal" together with $$T_1$$.  Remarkably characterized by Urysohn's Lemma and Tietze's Extension Theorem!
• 10/30: fall break
• 11/4: compactness: three characterizations.  Continuous images of compact sets are compact.  Closed subsets of compact sets are compact.  Continuous maps from compact spaces to $$T_2$$ spaces are closed.
• 11/6: Tychonoff's Theorem: a product of nonempty spaces is compact if and only if each factor is compact.  A subset $$A$$ of Euclidean space is compact if and only if it is closed and bounded.  Examples such as: spheres, $$\mathbb{CP}^n$$, $$\mathbb{RP}^n$$.
• 11/11: local compactness.  A Hausdorff space is locally compact iff every point has a compact neighborhood.  Topological manifolds are locally compact.  Continuous images of locally compact spaces under open maps are locally compact.
• 11/13: compactification.  One-point.  Or by taking the closure of the image of a space under an embedding.  Examples: affine curves compactified in projective space, locally compact $$T_2$$-spaces compactified in a cube (Stone-Čech compactification).
• 11/18: connectedness.  Definition, first properties.  Examples: $$I$$, $$\mathbb R$$; non-examples: discrete spaces, and the Sorgenfrey line.  Connectedness of subspaces.  Preserved by continuous images, like compactness, so quotients of connected spaces are connected.
• 11/20: proper maps (should have followed compactness).  Definition.  A useful special case: if $$Y$$ is a locally compact Hausdorff space, a map $$f\colon X\to Y$$ is proper if and only if it is closed and all fibres $$f^{-1}(y)$$ are compact.  Proper quotients of locally compact Hausdorff spaces are Hausdorff.  Application: proper group actions.
• 11/25: connected components are closed, but may or may not be open as well.  Totally disconnected spaces.  Path connectedness.
• 11/27: local versions: connected and locally path connected spaces are path connected.  Connected components of locally connected spaces are open.  That is equivalent to saying that the space of connected components $$\pi_0(X)$$ has the discrete topology when $$X$$ is locally connected.  Special case: if $$X$$ is compact as well as locally connected, it has only finitely many connected components.
• 12/2: homotopies and the compact-open topology.  Contractible spaces.  Retracts and deformation-retracts.  Definition of the fundamental group.

## Syllabus

Topology: beginning with the notions of open and closed sets and continuous functions familiar from analysis, we develop the general theory.  This allows us a very flexible notion of what a "space" is, compatible with constructions like cartesian products, quotients and subspaces.  We find the tools to deal with the notion of continuity throughout mathematics.  However, some of our old friends, like convergent sequences, may no longer behave as we expect, so we are led to consider a range of possible niceness conditions.  At the end of the course, we consider connectedness and the natural progression to algebraic topology.

Topics include:
• topological spaces
• convergence, separability and countability
• compactness
• connectedness
• metrization

## Assignments

Individual assignments are posted here; please note that late assignments will not be accepted.

## Exams

Midterm Solutions.