## Online lectures

Lectures are slide-shows consisting of videos (screencasts) mostly alternating with quizzes. You can make the videos full-screen. It is recommended that you watch with "quality" at least 720p to avoid fuzzy writing (click on the setting at the bottom of the YouTube video to specify this).

• Lecture 1: Introduction.

We begin with introductory remarks on topological properties and spaces. We then give a precise definition of topological spaces and give the simplest examples.

• Lecture 2: Some topological spaces.

We give examples of topological spaces, including the standard topology on $\mathbb{R}$.

• Lecture 3: Bases for topology.

A topology can be described using a basis. We define basis for a topology and give some examples.

• Lecture 4: Metric spaces, subspaces.

We define metric spaces and see some important examples.

• Lecture 5: Hausdorff distance, ultra-metrics

We introduce some important examples: the Hausdorff distance and the Cantor set viewed as an ultra-metric.

• Lecture 6: Subsets and points.

We define the subspace topology on a subset. We study the relation between subsets and points, defining interiors, closures and frontiers of sets.

• Lecture 7: Continuity.

We define continuity for functions between topological spaces and give the first examples.

• Lecture 8: Local criteria for continuity.

We characterize continuity in terms of conditions near each point, and in terms of neighbourhood bases. Using this, we study functions to $\mathbb{R}$.

• Lecture 9: More topological spaces and relations.

We describe initial and final topologies associated to collections of functions. We then describe isometric embeddings. In the final segment we study the Cantor set in detail and use it to construct space-filling curves.

• Lecture 10: Homeomorphisms.

We define Homeomorphisms and give many examples. We also describe the order topology associated to an ordered set.

• Lecture 11: Pasting and Fundamental covers.

The pasting lemma gives conditions for continuous functions to be glued together to continuous functions. We prove this and show the hypothesis holds for locally closed cover.

• Lecture 12: Connectedness.

We define connectedness, show various ways of deducing this and give examples. We show that any space decomposes into connected components.

• Lecture 13: Applications of connectedness.

We give applications of connectedness, in particular the intermediate value theorem and showing spaces are not homeomorphic in various examples.

• Lecture 14: Path connectedness.

We define path-connectedness and prove basic properties. We show that the topologists sine curve is connected but not path-connected.

• Lecture 15: Separation properties

We define various separation properties of topological spaces. We prove two deep results giving consequences of separation properties — Urysohn lemma and Tietze extension theorem.

• Lecture 16: Countability properties

We define countability properties – first and second countability and separability, of topological spaces. These are axioms that encapsulate a space not being extremely large. We derive some consequences.

• Lecture 17: Compactness

Compactness is the deepest property of topological spaces, with many important consequences. We define compactness, give criteria and consequences.

• Lecture 18: Compactness and metric spaces

We study consequences and criteria for compactness in the case of metric spaces. The most important of these is the relation between compactness and the existence of $\varepsilon$-nets.

• Lecture 19: Compactification

A compactification of a topological space $X$ is a compact topological space in which $X$ embeds as a dense subset. We introduce the one-point compactification. We define local compactness and give consequences of the one-point compactification for locally compact Hausdorff spaces.

• Lecture 20: Products of Spaces

We define the product of collections of topological spaces as a topological space. We prove many properties, in particular showing in what sense this is the correct definition.

• Lecture 21: Products, Metrization and Compactification

We study when products of metric spaces are metrizable, i.e., with topology the same as that induced by some metric. As a consequence of this and earlier results (Urysohn lemma) we prove the Urysohn metrization theorem. We also prove the Tychonoff theorem showing that arbitrary products of compact topological spaces is compact.

• Lecture 22: Baire category theorem

The Baire Category theorem is a deep result that allows one to speak of sets being small or large in a topological sense in metric spaces. We state and prove this. As an application we show that there exists a space that is regular but not normal.

• Lecture 23: Quotients of Spaces

The geometrically most important construction of topological spaces is that of quotient spaces, which are spaces obtained by taking quotients under an equivalence relation. We define the quotient topology, study its properties, and give a few examples.