Closed set on topological space

Question

This is a problem on topological spaces and continuous functions.

If $f,g \to\mathbb{R}$ are continuous functions, then $T=\{x\in X: f(x)=g(x)\}$ is closed on X

Exercise: Prove that this statement is also true for any other hausdorff space Y instead of $\mathbb R$ — Marm, Nov 27 '14 at 00:33

score 4 · Accepted Answer · answered Nov 27 '14 at 00:26

4

HINT: Prove that the function $f-g:X\to\Bbb R:x\mapsto f(x)-g(x)$ is continuous.

answered Nov 27 '14 at 00:26

Brian M. Scott

score 1 · Answer 2 · answered Nov 27 '14 at 00:26

1

$f-g$ is the difference of continuous functions, hence continuous. Then

$$(f-g)^{-1}(\{0\})$$

is the set in question and being the inverse image of a closed set, it is closed.

answered Nov 27 '14 at 00:26

Adam Hughes

2 Answers2