Simplicial Structures in Topology

I.1 Topology 13
Cx = �
Xj
j∈J
is a connected space that contains x, known as a connected component of x. The
space Cx is maximal in the sense that if a subset M ⊂ X is connected and contains Cx,
then Cx = M (in other words, the connected component Cx is the largest connected
subspace that contains x). Moreover, Cx is a closed subspace of X; in fact, by
Theorem (I.1.18), Cx is connected and since Cx ⊂ Cx, thenCx = Cx. The connected
components of two points x,y ∈ X are either disjoint or coincide: in fact, if z ∈
Cx ∩Cy, the subspace Cx ∪Cy is a connected subspace of X that contains both spaces
Cx and Cy; but the connected components are maximal and so Cx = Cx ∪Cy = Cy.
From this fact, we conclude that the relation “x,y ∈ X are in the same connected
component” is an equivalence relation. Therefore, a topological space X is a disjoint
union of maximal, closed, connected subspaces. A space is connected if and only if
it has only one connected component.
Finally, we note that two topological spaces with different numbers of connected
components cannot be homeomorphic. This is another criterion for verifying
whether two spaces are homeomorphic.
There is another type of connectedness, called path-connectedness. Apath in a
topological space X is a map f : [0,1] −→ X; two points x0,x1 ∈ X are joined by a
path if there is path f of X such that f (0)=x0 and f (1)=x1. We say that a space
X is path-connected if and only if any two points x0,x1 ∈ X maybejoinedbyapath
in X.
The results of Theorem (I.1.13) hold true for path-connectedness; as a matter of
completion (and to follow the preceding model), we present these results as a single
theorem.
(I.1.20) Theorem. The following statements are true:
1. Let f : X → Y be a continuous function, where X is path-connected; then the
space f (X) is path-connected.
2. Let {Xj | j ∈ J} be a set of path-connected subspaces of a space Y , with
�
j Xj �= /0;thenX= �
j Xj is a path-connected space.
3. If X and Y are path-connected, then X ×Y is path-connected.
Proof. 1. Given any two points y0,y1 of f (X), we choose x0,x1 ∈ X such that y0 =
f (x0) and y1 = f (x1). Because X is path-connected, there is a path g: [0,1] → X
such that g(0)=x0 and g(1)=x1. So,thepathfg: [0,1] → f (X) links y0 to y1 and
f (X) is, therefore, path-connected.
In particular, if a space X is path-connected, any space Y homeomorphic to X is
path-connected.
2. Given any two points x0,x1 ∈ X, suppose that x0 ∈ Xi 0 and x1 ∈ Xi 1 ;leta ∈
Xi 0 ∩ Xi 1 . By the hypothesis, there are two continuous functions
f0 : [0,1] → Xi0 , f0(0)=x0 , f0(1)=a,
f1 : [0,1] → Xi1 , f1(0)=x1 , f1(1)=a.