query: A space $X$ with property that all compact subspaces are closed.
positive_passages.docid: 147331
positive_passages.text: Compactness and Hausdorffness
negative_passages.docid: 8015
negative_passages.text: Compact set plus a closed set is a closed set
negative_passages.docid: 106957
negative_passages.text: Compact sets are closed?
negative_passages.docid: 125623
negative_passages.text: Subspaces of separable normed spaces
negative_passages.docid: 113534
negative_passages.text: Show that any metrizable space $X$ is Hausdorff
negative_passages.docid: 59654
negative_passages.text: Show that in a discrete metric space, every subset is both open and closed.
negative_passages.docid: 148213
negative_passages.text: Given a topological space $X$, why are both $X$ and $\emptyset$ open and closed?
negative_passages.docid: 148994
negative_passages.text: $K \subset \mathbb{R}^n$ is compact iff it is closed and bounded
negative_passages.docid: 47810
negative_passages.text: $C(X)$ is separable when $X$ is compact?
negative_passages.docid: 39076
negative_passages.text: Contraction of compact sets
negative_passages.docid: 78399
negative_passages.text: In $T_1$ space, all singleton sets are closed?
negative_passages.docid: 114608
negative_passages.text: Proof concerning regular space: there exist a closed set contained in any open set containing $x$
negative_passages.docid: 118765
negative_passages.text: A topological space is locally compact then here is an open base at each point has all of its set with compact closure
negative_passages.docid: 135157
negative_passages.text: Totally bounded, complete $\implies$ compact
negative_passages.docid: 128934
negative_passages.text: Every infinite Hausdorff space has an infinite discrete subspace
negative_passages.docid: 149220
negative_passages.text: If every compact set is closed, then is the space Hausdorff?
negative_passages.docid: 65358
negative_passages.text: Prove $X$ is compact iff for each family $\{ F_\alpha\}_{\alpha\in I}$ of closed subsets of $X$ with the FIP, we have $\cap F_\alpha\neq\varnothing$.