Simplicial Structures in Topology

16 I Fundamental Concepts
is a path on Sn joining a and b. In the event that b is antipodal to a, we choose any
other point c of Sn ; this point can be antipodal neither to a, nor to b. Therefore, with
the preceding method, we construct a path r1 on Sn which links a to c and then a
path r2 which links c to b; the function r : [0,1] → Sn ,definedby
�
r1(2t) 0 ≤ t ≤
r(t)=
1 2
r2(2 − 2t) 1
2 ≤ t ≤ 1
for every t ∈ [0,1], is a path from a to b (see Corollary (I.1.10)). Hence, S n is
path-connected (see also Fig. I.6).
I.1.3 Compactness
b
a
Fig. I.6
Let Y be a topological space. A covering of Y is a family U = {Uj | j ∈ J} of subsets
of Y such that Y = �
j∈J Uj. IfYis a subspace of X, a covering of Y by subsets of
X is a family U of subsets of X whose union contains Y. A covering U is finite if
the set J of the indexes is finite; U is an open covering if all its elements are open
in Y (or, for subsets of X, if they are open in X). A subcovering of U is a subset
U ′ = {U j ′ | j ′ ∈ J ′ } of U where J ′ ⊂ J. A covering U of Y is a refinement of a covering
V of Y if for every V ∈ V there exists U ∈ U such that U ⊂ V .
A topological space X is said to be compact if every open covering of X has a
finite subcovering; in other words, given any set U = {U j | j ∈ J} with Uj ⊂ X open
in X, foreveryj∈Jsuch that �
j Uj = X, there is a finite number of open sets Uj,
for instance, U1,U2,...,Un such that X = U1 ∪U2 ∪···∪Un. A subspace Y ⊂ X is
compact in the induced topology on Y if and only if every covering of Y by open sets
of X has a finite subcovering (because any open set U of Y is of the type U = V ∩Y ,
with V open in X).
The reader may easily prove that the space
X = {0}∪{1/n | n ∈ N} ,