This site is being phased out.
Connectedness
Redirect to:
Q: Can I travel anywhere I want in $X?$ If Yes, then $X$ is can be called "connected".
By travel from $A$ to $B$ we mean a continuous function $q: [0,1] → X$ such that $q(0)=A, q(1)=B$.
If this holds for all $A, B$ in a topological space $X$, then X is called, more precisely, path-connected.
Theorem. Suppose function $f: X → Y$ is continuous. Then, if $X$ path-connected then so is its image, $f(X)$, under $f$.
Proof. The proof is very simple. For any two points in $f(X)$ you need to be able to travel from one to the other. But in $f(X)$ every point comes from $X$ under $f$. So those two points have to be $f(A), f(B)$ for some $A, B$ in $X$. Now since $X$ is connected, there is a continuous function $q: [0,1] → X$ such that $q(0)=A, q(1)=B$. Then $fq: [0,1] → f(X)$ is continuous and $fq(0)=f(A), fq(1)=f(B)$. $\blacksquare$
The construction generates an equivalence relation on $X$. The equivalence classes are called connected components.
It is easy to show that connectedness is a topological invariant. And so is the number of components, see Betti numbers.
