query: Showing that $[0, \omega_1]$ is compact.
positive_passages.docid: 156978
positive_passages.text: First uncountable ordinal
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: 102048
negative_passages.text: Closure of a compact set in Regular space $ X $
negative_passages.docid: 97053
negative_passages.text: To prove $\omega_1 \cup \{\omega_1\} $ is not first countable but $\omega_1$ is
negative_passages.docid: 147331
negative_passages.text: Compactness and Hausdorffness
negative_passages.docid: 28637
negative_passages.text: Proving continuous image of compact sets are compact
negative_passages.docid: 103451
negative_passages.text: Show that $Y$ is locally compact.
negative_passages.docid: 100608
negative_passages.text: Prove that $\|v\|_{H^1(\Omega )}\leq C(\|f\|_{L^2(\Omega )}+\|v\|_{H^{1/2}(\partial \Omega )}+\|\partial _\nu v\|_{H^{1/2}(\partial \Omega )}))$
negative_passages.docid: 39076
negative_passages.text: Contraction of compact sets
negative_passages.docid: 21881
negative_passages.text: Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC
negative_passages.docid: 158498
negative_passages.text: $A\subset \mathbb{R}$ with more than one element and $A/ \{a\}$ is compact for a fixed $a\in A$
negative_passages.docid: 26563
negative_passages.text: Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
negative_passages.docid: 29303
negative_passages.text: Cartesian product of compact sets is compact
negative_passages.docid: 127173
negative_passages.text: Closed subset of compact set is compact
negative_passages.docid: 113696
negative_passages.text: Showing that the unit ball in $(C^\alpha[a, b], \|\cdot\|_\alpha)$ is compact in $(C[a, b], \|\cdot\|_\infty)$
negative_passages.docid: 148516
negative_passages.text: Proof that $\omega^\omega$ is completely metrizable and second countable
negative_passages.docid: 147213
negative_passages.text: Prove that finite unions of compact sets are compact