topology notes lboro

MAB298–Elements of Topology
May 20, 2021

Contents
1 Point set topology 2
1.1 Topological spaces and open sets . . . . . . . . . . . . 2
1.2 Topology on metric spaces, induced topology, closed sets 4
1.3 Interior, boundary and exterior of a set . . . . . . . . . 6
1.4 Closure, adherent, limit and isolated points . . . . . . . 9
1.5 Everywhere dense and nowhere dense sets . . . . . . . 11
1.6 Continuous maps . . . . . . . . . . . . . . . . . . . . . 12
1.7 Homeomorphism, topological properties . . . . . . . . . 16
1.8 Compactness ....................... 18
1.9 Properties of compact spaces . . . . . . . . . . . . . . . 20
1.10Connectedness....................... 22
1.11 Pathwise Connectedness . . . . . . . . . . . . . . . . . 24
1.12 Hausdorff spaces . . . . . . . . . . . . . . . . . . . . . 26
1.13 New spaces from old ones . . . . . . . . . . . . . . . . 29
2 Manifolds 33
2.1 Manifolds ......................... 33
2.2 Implicit Function Theorem . . . . . . . . . . . . . . . . 35
2.3 Surfaces as two-demsional manifolds . . . . . . . . . . 38
2.4 Closed surfaces and gluing operations . . . . . . . . . . 41
2.5 Classification of closed surfaces . . . . . . . . . . . . . 43
2.6 Figures........................... 46
3 The fundamental group 53
3.1 Homotopy......................... 53
3.2 Fundamental group . . . . . . . . . . . . . . . . . . . . 54
3.3 Coveringspaces...................... 55
3.4 The fundamental group of the circle . . . . . . . . . . . 56
1

Chapter 1
Point set topology
1.1 Topological spaces and open sets
Definition 1. Let Xbe a non-empty set. A collection τof subsets of
Xis said to be a topology on Xif
(i) Xand the empty set, ∅, belong to τ,
(ii) the union of any (finite and infinite) number of sets in τbelongs
to τ, and
(iii) the intersection of any two sets in τbelong to τ.
The pair (X, τ) is called a topological space. The elements of τare
called open sets.
Example 1.Let X={a, b, c, d, e}and
τ1={X, ∅,{a},{c, d},{a, c, d},{b, c, d, e}}.
Then τ1is a topology on Xas it satisfies conditions (i), (ii), (iii)
of Definition 1.
Example 2.Let X={a, b, c, d, e}and
τ2={X, ∅,{a},{c, d},{a, c, e},{b, c, d}}.
Then τ2is not a topology on Xas the intersection
{a, c, e}∩{b, c, d}={c}
of two members of τ2does not belong to τ2; that is, τ2does not satisfy
condition (iii) of Definition 1.
2

Example 3.Let Zbe the set of all integer numbers and let τconsists
of Z,∅, and all finite subsets of Z. Then τis not a topology on Z,
since the infinite union
{1}∪{2}∪{3} ∪ . . . {n} ∪ ··· =N
of members of τdoes not belong to τ; that is τdoes not have property
(ii) of Definition 1.
Any set Xcan be endowed with at least two different topologies
in the following natural way.
Definition 2. Let Xbe a non-empty set and let τbe the collection
of all subsets of X.1Then τis called the discrete topology on the set
X. The topological space (X, τ ) is called a discrete space. In other
words, Xis discrete iff every A⊂Xis open.
We note that τin Definition 2 does satisfy the conditions of Defi-
nition 1 and so is indeed a topology.
Definition 3. Let Xbe any non-empty set and let τ={X, ∅}. Then
τis called the indiscrete topology on the set Xand (X, τ) is said to be
an indiscrete space. In other words, Xis indiscrete iff the only open
sets in Xare the empty set and Xitself.
Once again we have to check that τsatisfies the conditions of
Definition 1 and so is indeed a topology.
Example 4.The standard topology on the real line Ris defined in the
following way: a subset A⊂Ris open if and only if for any x∈A
there is δ > 0 such that (x−δ, x +δ)⊂A. Similarly, one defines the
topology in Rn: a subset A⊂Rnis open if and only if for any x∈A
there is δ > 0 such that Bδ(x)⊂A, where
Bδ(x) = {y∈Rn| |x−y|< δ}
is the open ball of radius δcentered at x.
The definition of a topological space is very general. There are
many examples of topological spaces with very different and sometimes
exotic properties.
1τis called the power set of X, denoted P(X) or 2X.
3

Example 5.On the real line R, we define the topology τby introducing
the following open subsets: R, empty set ∅, and infinite intervals of the
form (a, +∞), where a∈R. To prove that this collection of subsets
satisfies all necessary properties it suffice to notice that
1) (a1,+∞)∩(a2,+∞) = (a, +∞)∈τ, where a= max{a1, a2},
and
2) ∪α∈I(aα,+∞) = (a, +∞)∈τ, where a= infα∈I{aα}.
1.2 Topology on metric spaces, induced
topology, closed sets
Rand Rnare just two particular cases of the following general con-
struction allowing us to introduce ”topology” on any metric space
(X, d).
Recall the following standard definition.
Definition 4. A metric space is a pair (X, d) where Xis a set, and
dis a metric on X, that is a function from X×Xto Rthat satisfies
the following properties for all x, y, z ∈X:
d(x, y)≥0, and d(x, y) = 0 if and only if x=y,
d(x, y) = d(y, x) (symmetry), and
d(x, y)≤d(x, z) + d(z, y) (triangle inequality).
A metric space (X, d) possesses the following natural topology.
Definition 5. A set Uin a metric space (X, d) is open if and only if
for any x∈Uthere exists δ > 0 such that Bδ(x)⊂U, where
Bδ(x) = {y∈X:d(y, x)< δ}
is the open ball of radius δcentered at x.
Conclusion: Every metric space is a topological space.
Obviously, the topology of Rnis a particular case of this construc-
tion: d(x, y) = |x−y|=pP(xi−yi)2for x, y ∈Rn.
It is very important that the above construction allows us to intro-
duce topology on infinite-dimensional spaces. Recall two examples:
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