Approximating a characteristic function of an open set using a continuous function

Question

I am in a middle of solving a problem and I am looking for a way to approximate a characteristic function of an open set using a continuous function. To be more specific I am looking for a function in $C(X)$ (where X is compact metric space) that will get me close to $\chi_A$ where $A$ is open. In fact, the approximation can be terrible as long as the function is $0$ outside $A$ and some positive value inside $A$. I noticed online that you can use Urysohn Lemma but I would avoid that. Luzin's theorem would be of help if I was over an interval, but I am not.

any hints would be appreciated.

Thank you!

Answer 1

You can use a bump function:

https://en.wikipedia.org/wiki/Bump_function

Alternatively, you can use a Tychonoff embedding to send your compact set to a cube: $[0,1]^J$ and use Luzin's from there.