query: Existence of a closed and open set in 0-dimensional Hausdorff space
positive_passages.docid: 144731
positive_passages.text: Disjoint closed sets in a second countable zero-dimensional space can be separated by a clopen set
negative_passages.docid: 80860
negative_passages.text: How to call a space in which every G$_\kappa$-set is open
negative_passages.docid: 119105
negative_passages.text: If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
negative_passages.docid: 30079
negative_passages.text: Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?
negative_passages.docid: 73130
negative_passages.text: A simple question about open set
negative_passages.docid: 27998
negative_passages.text: Prove existence of disjoint open sets containing disjoint closed sets in a topology induced by a metric.
negative_passages.docid: 35573
negative_passages.text: A closed ball in a metric space is a closed set
negative_passages.docid: 41611
negative_passages.text: Open and closed set in n by n matrices space.
negative_passages.docid: 49220
negative_passages.text: Open Sets of $\mathbb{R}^1$ and axiom of choice
negative_passages.docid: 128730
negative_passages.text: Open subspaces of locally compact Hausdorff spaces are locally compact
negative_passages.docid: 2915
negative_passages.text: $X$ is Hausdorff if and only if the diagonal of $X\times X$ is closed
negative_passages.docid: 127173
negative_passages.text: Closed subset of compact set is compact
negative_passages.docid: 43535
negative_passages.text: show that the interval of the form $[0,a)$ or $(a, 1]$ is open set in metric subspace $[0,1]$ but not open in $\mathbb R^1$
negative_passages.docid: 3023
negative_passages.text: Construction of a Borel set with positive but not full measure in each interval
negative_passages.docid: 127900
negative_passages.text: Interior of closure of open set in topological space
negative_passages.docid: 72468
negative_passages.text: Open and closed balls in $C[a,b]$
negative_passages.docid: 42494
negative_passages.text: The boundary is a closed set