Talk:Intersection of Closed Set with Compact Subspace is Compact/Proof 1

Bltzmnn.k, it seems you wanted to address the ambiguity between compactness of a space and compactness in an ambient space. But I'm afraid your edit introduces many errors:


 * The definition of the topology of $T_K$ is indirect. You just want to say that $\tau_K$ is the induced topology on $K$, but I see no need to do so.
 * The conclusion that $H\cap K$ is a union of open sets cannot be correct.

Please let us know what you think could have been improved, because it is unclear. If you want to address the fact that the above notions of compactness are equivalent (which they are), this is not the place to do so. See Equivalent Definitions of Compact Topological Subspace. --barto (talk) 13:12, 30 August 2017 (EDT)
 * I personally don't like the way the definitions of a compact subspace are currently presented. It needs to be stressed that they are equivalent. As it stands, pages such as Definition:Compact seem to imply that they are not. --barto (talk) 13:16, 30 August 2017 (EDT)