I need to show that: All zero dimensional spaces are completely regular.
Here are my definitions: Recall that a space is called zero dimensional if each point has a neighborhood base consisting of sets which are both open and closed. In particular the Michael line M, the Sorgenfrey line S, and the countable ordinals w_1 are all completely regular spaces.
Note: We say that a point x ∈X has a neighborhood base at x if there is a collection {Uxj: j=1-> ∞} of open subsets of X such that every neighborhood W of x contains some Uxj.
Definition of completely regular space
A completely regular space is a topological space in which, for every point and a closed set not containing the point, there is a continuous function that has value 0 at the given point and value 1 at each point in the closed set.
I was thinking to prove it by contradiction.
So here I go:
Suppose X is not completely regular then there exists one point and every closed set not containing the point,s.t. there is not a continuous function that has value 0 at the given point and value 1 at each point in the closed set. then what?!
Please help.
Best Answer
HINT: Don’t make it too hard; a direct proof is easy. If $H$ is a clopen subset of a space $X$, then the map
$$f:X\to\Bbb R:x\mapsto\begin{cases}0,&\text{if }x\in H\\1,&\text{if }x\in X\setminus H\end{cases}$$
is continuous. Prove this, and you’re almost done.