MA342R - Covering Spaces and Fundamental Groups Dr. David R. Wilkins Lecture Slides

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Details of Lecture Material

Lecture 1 (January 16, 2017)

This was an introductory lecture, delivered extempore

Basic Results concerning Topological Spaces

Lecture 2 (January 19, 2017)

The lecture began with the axioms characterizing the properties of the collection of open sets in a topological space. This lecture reviewed basic definitions and results concerning topological spaces. This included some discussion of metric spaces. In particular the definition of open sets in metric spaces was presented, open balls in metric spaces were shown to be open sets (Lemma 1.1). and the collection of open sets in a metric space was shown to satisfy the topological space axioms (Proposition 1.2). The lecture continued by developing the general theory of topological spaces, discussing in particular the following topics: the definition of closed sets in a topological space; the basic properties of the collection of closed sets in a topological space (Proposition 1.3); neighbourhoods of points in a topological space; interiors and closures of subsets of a topological space. It was shown that an open set is disjoint from the closure of a subset of a topological space if and only if it is disjoint from the subset itself (Lemma 1.6). Also the complement of the closure of a subset of a topological space was shown to be the interior of the complement of that subset (Proposition 1.7).

Lecture 3 (January 20, 2017)

The lecture began by showing that the definition of neighbourhoods in general topological spaces is consistent with a standard definition of neighbourhoods of points in metric spaces that employs the distance function on a metric space. Much of the remainder of the lecture was concerned with subspace topologies. In particular it was shown that, on any subset of a metric space, the subspace topology coincides with the metric space topology (Corollary 1.12). The lecture concluded by giving the definition of Hausdorff spaces, proving that metric spaces are Hausdorff spaces (Lemma 1.17), and showing also that infinite sets with the cofinite topology are not Hausdorff spaces. The lecture concluded with brief informal discussion of the Zariski topology on ℝⁿ (not included in the course notes for MA342R in Hilary Term 2017). A subset of ℝ is closed with respect to the Zariski topology on ℝ if and only if it is the set of common zeros of some collection of polynomials in n indeterminates with real coefficients. (A fuller discussion of Zariski topologies on algebraic sets is to be found in Section 11 of the course notes for TCD Mathematics module MA3412 in Hilary Term 2010.)

Lecture 4 (January 23, 2017)

This lecture discussed continuous functions between topological spaces. A function f : X → Y from a topological space X to a topological space Y is continuous if and only if the preimage f^-1(V) of every open set V in Y is open in X. The lecture introduced the definition of continuity at a point of a topological space: a function f : X → Y from a topological space X to a topological space Y is continuous at a point p of X if and only if the preimage f^-1(N) of every neighbourhood N of f(p) in Y is a neighbourhood of p in X. It was proved that a function f : X → Y is continuous on X if and only if it is continuous at every point of X (see Proposition 1.20; see also Proposition 1.6.15 of Mathematics MA3421, Functional Analysis I, Chapter 1 (Michaelmas Term 2016, TCD, R.M. Timoney, TCD) ). The lecture also included a statement and proof of the Pasting Lemma (also known as the Gluing Lemma): if a topological space is represented as a finite union of closed subsets, then a function mapping that space to a topological space is continuous if and only if its restriction to each of those closed subsets is continuous (see Lemma 1.24). (Material from Proposition 1.25 onwards was not discussed on January 23, 2017, but was subsequently covered at the beginning of the later lecture on January 30, 2017.)

Lecture 5 (January 26, 2017)

This lecture discussed product and quotient topologies. The lecture began with a discussion of bases for topologies. A collection β of subsets of a topological space X is a base for the topology of X if the open sets of X are those subsets of X that are unions of sets belonging to the base β. There are necessary and sufficient conditions that a collection β of subsets of X must satisfy in order that β be a base for a topology on X (see Proposition 1.26; see also Proposition 1.4.11 of Mathematics MA3421, Functional Analysis I, Chapter 1 (Michaelmas Term 2016, TCD, R.M. Timoney, TCD) ). The product topology on a Cartesian product of topological spaces is the topology generated by a base consisting of products of open sets in the topological spaces out of which the Cartesian product is formed. It was shown that a continuous map from a topological space into a finite product of topological spaces is continuous if and only if all its components are continuous (see Proposition 1.30); see also Proposition 1.5.13 of Mathematics MA3421, Functional Analysis I, Chapter 1 (Michaelmas Term 2016, TCD, R.M. Timoney, TCD) ). The product topology on ℝⁿ was proved to be the same as the topology generated by the Euclidean distance function on ℝⁿ (see Proposition 1.31). The lecture concluded with a discussion of identification maps and quotient topologies.

Lecture 6 (January 27, 2017)

This lecture reviewed properties of compact topological spaces. In particular the following standard results were obtained: the one-dimensional Heine-Borel Theorem, which asserts that closed bounded intervals are compact (Theorem 1.37); a continuous function maps compact sets to compact sets (Lemma 1.39); a continuous real-valued function on a compact topological space attains maximum and minimum values on that space (Proposition 1.41); a compact subset of a Hausdorff space is closed (Corollary 1.43); a continuous bijection from a compact topological space to a Hausdorff space is a homeomorphism (Theorem 1.45); a continuous surjection from a compact topological space to a Hausdorff space is an identification map (Proposition 1.46). The lecture concluded with a proof of the Lebesgue Lemma (Lemma 1.47). The Lebesgue Lemma was then applied to prove that any continuous function from a compact metric space to a metric space is uniformly continuous (see Theorem 1.48).

Lecture 7 (January 30, 2017)

The lecture began by discussing some results included in the lecture notes for Lecture 4 that had not been covered on January 23. The pointwise definition of continuity for functions between topological spaces was shown to be consistent with the standard definition for functions between metric spaces (see Proposition 1.25). The lecture also discussed homeomorphisms between topological spaces. The lecture then proceeded with a discussion of products of compact topological spaces. It was shown that the product of a finite number of compact topological spaces is compact (Theorem 1.49). Various previous results were then combined to show that a subset of ℝⁿ is compact if and only if it is both closed and bounded (Theorem 1.50). Lecture 7 (January 30, 2017)

Lecture 8 (February 2, 2017)

This lecture reviewed the definition and basic properties of connectedness. The following standard results were obtained: a topological space is connected if and only if every continuous function mapping that space to the set of integers is constant (Corollary 1.56); intervals in the real line are connected (Theorem 1.57); every continuous integer-valued function defined on an interval in the real line is constant (Corollary 1.58); the closure of a connected set is connected (Lemma 1.60); continuous functions map connected sets to connected sets (Lemma 1.61); the product of two connected topological spaces is connected (Lemma 1.62). It was shown that any topological space can be expressed as the disjoint union of its connected components (Proposition 1.63).

Lecture 9 (February 3, 2017)

The lecture introduced the notion of path-connectedness, and proved that every path-connected topological space is connected (Proposition 1.64). The lecture included an example of a connected topological space that is not path-connected. The lecture continued with definitions of locally connected and locally path-connected topological spaces. It was shown that all connected, locally path-connected topological spaces are path-connected (Proposition 1.66). The definition of a contractible topological space was given, and it was shown that all contractible topological spaces are path-connected (Lemma 1.69). The Comb Space was discussed. This is an example of a subset of the plane that is contractible but not locally connected.

Winding Numbers of Closed Curves in the Plane

Lecture 10 (February 6, 2017)

Lecture 11 (February 9, 2017)

Lecture 12 (February 10, 2017)

The Fundamental Group of a Topological Space

Lecture 13 (February 13, 2017)

Lecture 14 (February 16, 2017)

Lecture 15 (February 17, 2017)

Covering Maps

Lecture 16 (February 20, 2017)

Lecture 17 (February 23, 2017)

Lecture 18 (February 24, 2017)

Free Discontinuous Group Actions on Topological Spaces

Lecture 19 (March 6, 2017)

Lecture 20 (March 9, 2017)

Lecture 21 (March 10, 2017)

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Dr. David R. Wilkins, School of Mathematics, Trinity College Dublin.