query: Prove that if a space $X$ is locally path connected then each path component of each open subset of $X$ is open.
positive_passages.docid: 157830
positive_passages.text: Definition of locally pathwise connected
negative_passages.docid: 39746
negative_passages.text: How to prove that given set is a connected subset of the space of matrices?
negative_passages.docid: 127365
negative_passages.text: Question on showing a bijection between $\pi_1(X,x_0)$ and $[S^1, X]$ when X is path connected.
negative_passages.docid: 35631
negative_passages.text: $H_0(X,A) = 0 \iff A \cap X_i \neq \emptyset \forall $ path-components $X_i$
negative_passages.docid: 108394
negative_passages.text: Proof of that every interval is connected
negative_passages.docid: 153492
negative_passages.text: Let $p\colon X \to Y$ be a quotient map then if each set $p^{-1}({y})$ is connected, and if $Y$ is connected, then $X$ is connected.
negative_passages.docid: 108052
negative_passages.text: Does any connected metrizable space with more than one element have uncountably many open sets, each generating a connected subspace?
negative_passages.docid: 102084
negative_passages.text: How to show path-connectedness of $GL(n,\mathbb{C})$
negative_passages.docid: 42811
negative_passages.text: Product of path connected spaces is path connected
negative_passages.docid: 116009
negative_passages.text: Proving that if a set is both open and closed then it is equal to the real numbers
negative_passages.docid: 15480
negative_passages.text: Open set in a metric space is union of closed sets
negative_passages.docid: 126192
negative_passages.text: Connected, locally connected, path-connected but not locally path-connected subspace of the plane
negative_passages.docid: 17470
negative_passages.text: Does path-connected imply simple path-connected?
negative_passages.docid: 34177
negative_passages.text: Prove that $\mathbb{R}^2\setminus\mathbb{Q}^2$ is connected without path connectedness
negative_passages.docid: 135783
negative_passages.text: Let $T$ and $T^{'}$ be topologies on set $X$ such that $T \leq T^{'}$. If $(X,T^{'})$ is path-connected, is $(X,T)$ path connected?
negative_passages.docid: 146789
negative_passages.text: A locally constant sheaf on a locally connected space is a covering space; Proof?
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