query: Suppose a subspace of $X$ is compact iff closed. Then is $X$ compact hausdorff?
positive_passages.docid: 147331
positive_passages.text: Compactness and Hausdorffness
negative_passages.docid: 66334
negative_passages.text: How is every subset of the set of reals with the finite complement topology compact?
negative_passages.docid: 86317
negative_passages.text: Show that $F\subset Y$ is closed in $Y$ iff $F=Y\;\cap\;H$ where $H\subset X$ is closed in $X$.
negative_passages.docid: 148403
negative_passages.text: Compact space and Hausdorff space
negative_passages.docid: 86782
negative_passages.text: A $T_4$ space satisfying that 'every continuous map from it to a Hausdorff space is a closed map' is compact.
negative_passages.docid: 71916
negative_passages.text: Quasi-components and components coincide for compact Hausdorff spaces
negative_passages.docid: 99645
negative_passages.text: Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
negative_passages.docid: 113534
negative_passages.text: Show that any metrizable space $X$ is Hausdorff
negative_passages.docid: 106957
negative_passages.text: Compact sets are closed?
negative_passages.docid: 73353
negative_passages.text: Is it true that a subset that is closed in a closed subspace of a topological space is closed in the whole space?
negative_passages.docid: 14282
negative_passages.text: $X/{\sim}$ is Hausdorff if and only if $\sim$ is closed in $X \times X$
negative_passages.docid: 11383
negative_passages.text: Disjoint compact sets in a Hausdorff space can be separated
negative_passages.docid: 58948
negative_passages.text: If $A$ is compact, is then $f(A)$ compact?
negative_passages.docid: 3708
negative_passages.text: How can I prove that compact subset iff sequentially compact in second countable topological spaces?
negative_passages.docid: 65180
negative_passages.text: A compact Hausdorff space that is not metrizable
negative_passages.docid: 100121
negative_passages.text: Assume that $(\text{X}, T)$ is compact and Hausdorff. Prove that a comparable but different topological space $(\text{X},T')$ is not.
negative_passages.docid: 29303
negative_passages.text: Cartesian product of compact sets is compact