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I am using the definition of compactly generated space from The Category of CGWH Spaces, which is

In $\mathbf{Top}$, a $k$-closed subset $Y\subset X$ is a set such that $u^{-1}(Y)$ is closed in $C$ for any $u: C\to X$ where $C$ is compact Hausdorff.

A space is compactly generated if all $k$-closed subsets are closed.

locally compact means every point has a local base of compact sets.

This is different from the definition from Wikipedia

So, is any compact space compactly generated?

And, is any locally compact space compactly generated?

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For your first question, if $X$ is compact Hausdorff, consider taking $C=X$ and $u$ the identity. –  Nate Eldredge May 6 at 5:12
    
Yes, I know this. But I want to know what about non-Hausdorff space –  Minghao Liu May 6 at 5:59
    
The term compactly generated is used differently by different authors. Sometimes is denotes a space where a subset is closed if it intersects each compact subset in a closed set, let's call these c-spaces. Other times it denotes a space as you describe it, but these are also called k-spaces. Both compact and locally compact spaces are c-spaces. So you are actually searching for a space that is a c-space but not a k-space. –  Stefan Hamcke May 19 at 14:48
    
It seems that both your question and this require a space with many compact subsets which are not images of compact Hausdorff spaces. –  Stefan Hamcke May 19 at 14:49

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