Simplicial Structures in Topology

I.1 Topology 9
that f is a homeomorphism between X and Y. A homeomorphism is really a 1-1
correspondence between points of X and Y which induces an 1-1 correspondence
between their respective topologies (that is to say, between the open sets of X and
the open sets of Y ). In practice we make no distinction between two homeomorphic
spaces. Let q: X → Y be a surjection from a space X onto a space Y with the
quotient topology induced by q. Then the function q is continuous; the function
q: X → Y is called quotient map.
The next result provides a link between homeomorphisms and quotient spaces.
(I.1.11) Lemma. Let f : X → Y be a homeomorphism and let ≡X, ≡Y be equivalence
relations in X and Y , respectively. Then X/ ≡X and Y / ≡Y are homeomorphic,
provided that
x ≡X x ′ ⇐⇒ f (x) ≡Y f (x ′ ).
Proof. Consider the following commutative diagram (that is to say, such that
FqX = qY f )
qX
X
��
X /≡X
f
F
��
Y
qY
��
��
Y /≡Y
where F is defined as follows: for each [x] ∈ X /≡X , F([x]) :=[f (x)]. F is a function:
if x ≡X x ′ ,thenf (x) ≡Y f (x ′ ) and therefore the entire class [x] is transformed univocally
into class [ f (x)]. Since the composite function qY f is continuous, so is FqX;
but the space X/ ≡X has the quotient topology and therefore (see and do Exercise 7
on p. 27) F is continuous.
At this point, let us consider the inverse function f −1 : Y → X and, as in the case
of f , let us construct the function
F ′ : Y /≡Y → X /≡X , F′ ([y]) :=[f −1 (y)]
for [y] ∈ Y/ ≡Y .AlsoF ′ is a function, for the same reason given for F.
The function F ′ defined above is also continuous and
F ′ F = 1 Y/≡Y , FF′ = 1 X/≡X
in other words, F is a homeomorphism. �
I.1.2 Connectedness
(I.1.12) Theorem. Let X be a topological space. The following statements are
equivalent:
(i) The empty set /0 and the set X itself are the only two subsets of X that are both
open and closed.