# Continous function

• Mar 15th 2009, 01:20 PM
math_help
Continous function
Let f: (X1,d1) -> (X2, d2) be a continous function. Let C, a subset of X2 be compact.

Is f^-1(C) ("f inverse") compact in X2?

Having trouble finding a counterexample...
• Mar 15th 2009, 02:50 PM
TheEmptySet
Quote:

Originally Posted by math_help
Let f: (X1,d1) -> (X2, d2) be a continous function. Let C, a subset of X2 be compact.

Is f^-1(C) ("f inverse") compact in X2?

Having trouble finding a counterexample...

I'm assuming that you mean $X_1$

Consider the function

$f: \mathbb{R}\to \mathbb{R}$ with the usual metric

$f(x)=1$

The one point set $\{ 1 \}$ is compact in the range. (It is closed and bounded)

but $f^{-1}(1)=\mathbb{R}$ this set is unbouned and not compact. :)
• Mar 16th 2009, 11:26 AM
math_help
What do you mean by saying I assume you mean X1?
• Mar 16th 2009, 11:31 AM
TheEmptySet
Quote:

Originally Posted by math_help
What do you mean by saying I assume you mean X1?

Quote:

Is f^-1(C) ("f inverse") compact in X2?

$f^{-1}(C) \subset X_1$ not $X_2$