query: How to find that the following set is connected and compact or not?
positive_passages.docid: 146455
positive_passages.text: Compactness of the set of all unitary matrices in $M_2(\mathbb{C})$
negative_passages.docid: 89264
negative_passages.text: Connectedness in the Quotient Space
negative_passages.docid: 22464
negative_passages.text: Which of the following subsets of   $\mathbb{R}^2$ are compact
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: 26563
negative_passages.text: Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
negative_passages.docid: 133633
negative_passages.text: A question about path-connected and arcwise-connected spaces
negative_passages.docid: 61930
negative_passages.text: Intersection of Closed and Compact Set is Closed
negative_passages.docid: 23225
negative_passages.text: Union of connected subsets is connected if intersection is nonempty
negative_passages.docid: 102484
negative_passages.text: Connectedness problem and countability of compact subsets of the Sorgenfrey line
negative_passages.docid: 35069
negative_passages.text: Connectedness of the boundary
negative_passages.docid: 62911
negative_passages.text: Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected
negative_passages.docid: 80524
negative_passages.text: Path-connected and locally connected space that is not locally path-connected
negative_passages.docid: 72556
negative_passages.text: Is the closure of path-connected set path connected?
negative_passages.docid: 9268
negative_passages.text: Proving that the Union of Two Compact Sets is Compact
negative_passages.docid: 44673
negative_passages.text: $M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
negative_passages.docid: 136108
negative_passages.text: Arbitrary intersection of closed, connected subsets of a compact space connected?
negative_passages.docid: 158189
negative_passages.text: If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected