I think this holds if $X$ is a CW complex. I don't have a reference, but try Bredon or Iversen.

I think this holds if $X$ is a CW complex. I don't have a reference, but try Bredon or Iversen.

Edit: Bredon, "Sheaf Theory", Chapter III "Comparison with Other Cohomology Theories". Perhaps this contains what you are looking for.

I think this holds if $X$ is a CW complex. I don't have a reference, but try Bredon.

I think this holds if $X$ is a CW complex. I don't have a reference, but try Bredon or Iversen.