Simplicial Structures in Topology

I.1 Topology 21
is the diagonal of X ×X,thenX is Hausdorff if and only if the diagonal ΔX is closed
in X ×X. We now apply the previous corollary with X = Y and W =(X × X)� ΔX.
It follows from the hypothesis B ∩C = /0thatB ×C ⊂ W. �
(I.1.35) Corollary. Let U be an open set of a compact Hausdorff space X. For
every x ∈ U there is an open set V ⊂ X such that
where V is compact.
x ∈ V ⊂ V ⊂ U
Proof. The spaces {x} and X � U are disjoint and closed in X; hence by
Theorem (I.1.23), they are compact. By the previous corollary we can find two
disjoint open sets V and W of X where x ∈ V, X �U ⊂ W. And so, we may conclude
that V ⊂ U. On the other hand, X �W is a closed set that contains V and is
contained in U; then, as we wished to prove, V ⊂ U. The compactness of V follows
from Theorem (I.1.23). �
The compact spaces of Rn are special. Before the next theorem, we give this
definition: X ⊂ Rn is bounded if there is an R > 0andann-disk Dn R such that
⊃ X.
D n R
(I.1.36) Theorem. A subset X ⊂ R n is compact if and only if X is closed and
bounded.
Proof. ⇒: By Theorem (I.1.25), X is closed. In order to prove that X is bounded,
let us take a real number ε > 0 and the covering
A = {D n ε(x) | x ∈ X}
of X. Since X is compact, there exists a finite subcovering of A that contains X;
suppose, for instance, that
X ⊂ D n ε(x1) ∪···∪D n ε(xr).
Then given any x0 ∈ X, the disk Dn 2rε (x0) contains X.
⇐: The space X is contained in an n-disk Dn R of Rn ;butDn R is contained in a
hypercube Cn of Rn whose edges are homeomorphic to I and so Cn is compact (see
Theorems (I.1.30) and (I.1.31)). Therefore X is a closed subspace of a compact
space and, by Theorem (I.1.25), it is compact. �
We now are able to identify the quotient spaces defined in Examples (I.1.6)
and (I.1.7) with more familiar topological spaces. Remember that the torus T 2 is
the space S1 × S1 with the product topology. Let f : I2 ≡ → T 2 be the function that
takes each equivalence class [s,t] ∈ I2 ≡ to (e2πis ,e2πit ) ∈ T 2 . This is a bijective, con-
tinuous function. On the other hand, I 2 ≡
is compact (it is the image of the compact
space I 2 by the quotient map) and T 2 is Hausdorff; therefore, by Theorem (I.1.27),
f is a homeomorphism.