Topology Videos

Below you can find Youtube links for my topology lecture videos. The videos can be used both for the advanced undergraduate course Math 431, and for the graduate core course Math 631, even though the undergraduate course might not cover all topics, while the graduate course might cover more. The videos are based on my lecture notes, and the book section numbers refer to the book Topology by James Munkres (Second edition, Pearson Modern Classics, 2017).

	Topic	Book Sections

I.	Topological Spaces

I.1	What is Topology?
I.2	Topological Spaces	12
I.3	Basis for a Topology	13
I.4	Topology via Order and Products	14, 15
I.5	The Subspace Topology	16
I.6	Closed Sets and Limit Points	17
I.7	Limits of Sequences and Separation Axioms	17
I.8	The Metric Topology	20

II.	Continuity of Functions

II.1	Continuous Functions	18
II.2	Topology of Infinite Products	19
II.3	Continuity in Metric Spaces	20, 21
II.4	The Quotient Topology	22

III.	Connectedness and Compactness

III.1	Connected Spaces	23
III.2	Connected Subspaces of the Real Line	24
III.3	Components and Local Connectedness	25
III.4	Compact Spaces	26, 27
III.5	Products of Compact Spaces	26, 37
III.6	Compactness in the Reals and Metric Spaces	27, 28
III.7	Local Compactness	29

IV.	Countability and Separation Axioms

IV.1	The Countability Axioms	30
IV.2	More Separation Axioms	31, 32
IV.3	The Urysohn Lemma	33
IV.4	The Urysohn Metrization Theorem	34
IV.5	The Tietze Extension Theorem	35

V.	Fundamental Group and Covering Spaces

V.1	Homotopy of Paths	51
V.2	Some Terminology from Group Theory	52
V.3	The Fundamental Group	52, 59, 60
V.4	Covering Spaces and Liftings	53, 54
V.5	A Sampling of Fundamental Groups	54, 59, 60
V.6	Higher Homotopy Groups and Then?