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

Links to Lecture Slides
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
ℝ^{n} (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 ℝ^{n} was proved
to be the same as the topology generated by the Euclidean
distance function on
ℝ^{n}
(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 onedimensional HeineBorel 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 realvalued 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 ℝ^{n} 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 integervalued 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
pathconnectedness, and proved that
every pathconnected topological space is connected
(Proposition 1.64).
The lecture included an
example
of a connected topological space that is not pathconnected.
The lecture continued with definitions of
locally connected
and
locally pathconnected
topological spaces. It was shown that all connected,
locally pathconnected topological spaces are
pathconnected
(Proposition 1.66).
The definition of a
contractible
topological space was given, and it was shown that
all contractible topological spaces are pathconnected
(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)
Back to D.R. Wilkins: MA342R
Back to D.R. Wilkins: Lecture Notes
Dr. David R. Wilkins,
School of Mathematics,
Trinity College Dublin.