query: Prove that $Y$ is compact if and only if $X$ is compact
positive_passages.docid: 157578
positive_passages.text: perfect map in topology
negative_passages.docid: 139537
negative_passages.text: Show that $x\subseteq y \Leftrightarrow \mathcal{P}(x)\subseteq\mathcal{P}(y)$.
negative_passages.docid: 133152
negative_passages.text: (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
negative_passages.docid: 96153
negative_passages.text: $ x^3<y^3 \iff x<y $ proof issue.
negative_passages.docid: 102048
negative_passages.text: Closure of a compact set in Regular space $ X $
negative_passages.docid: 128457
negative_passages.text: Show that $T$ is a compact operator
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 $\cap F_\alpha\neq\varnothing$.
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 83311
negative_passages.text: Compactness of $A:=${$f \in C[0,1], |f|_\infty \le K, |f'|_\infty \le M$}
negative_passages.docid: 72920
negative_passages.text: $x \perp y$ if and only if $\Vert x + \alpha y \Vert \ge \Vert x \Vert$ for all scalars $\alpha$
negative_passages.docid: 127173
negative_passages.text: Closed subset of compact set is compact
negative_passages.docid: 8015
negative_passages.text: Compact set plus a closed set is a closed set
negative_passages.docid: 47903
negative_passages.text: How to prove every closed interval in R is compact?
negative_passages.docid: 29679
negative_passages.text: Prove: $|x-y|\leq |x|+|y|$
negative_passages.docid: 110744
negative_passages.text: How to prove that a compact set in a Hausdorff topological space is closed?
negative_passages.docid: 51914
negative_passages.text: Show that $S$ is compact.
negative_passages.docid: 19111
negative_passages.text: If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.