Compact Space already CG Space

Question

A topological space $X$ is compactly generated if it's topology is determined as follows:

$U \subset X$ open if and only if $s^{-1}U \subset K$ open for all locally compact spaces $K$ and continuous $s:K \to X$.

I want to know if EVERY compact space is compactly generated.

Answer 1

Suppose $X$ is compact .

We want to show $U$ open iff $s^{-1}[U]$ open in $K$ for all continuous $s: K \to X$.

The left to right implication always holds (it's the definition of continuity of $s$). Suppose then that $U$ satisfies the right hand side condition. In particular we can take $K = X$ as $X$ is compact and $s(x) = x$. By assumption $s^{-1}[U] = U$ is open in $K = X$ so we are done.

Answer 2

Hint In a compact space $X$ you have $V$ is open $\Leftrightarrow X \backslash V$ is closed $\Leftrightarrow X \backslash V$ is compact.

Also, $s^{-1}(X \backslash V)=Y\backslash s^{-1}(V)$ for all functions $s : X \to Y$.