Category:Path-Connected Sets
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This category contains results about Path-Connected Sets in the context of Topology.
Let $T = \struct {S, \tau}$ be a topological space.
Let $U \subseteq S$ be a subset of $S$.
Let $T' = \struct {U, \tau_U}$ be the subspace of $T$ induced by $U$.
Then $U$ is a path-connected set in $T$ if and only if every two points in $U$ are path-connected in $T\,'$.
That is, $U$ is a path-connected set in $T$ if and only if:
- for every $x, y \in U$, there exists a continuous mapping $f: \closedint 0 1 \to U$ such that:
- $\map f 0 = x$
- and:
- $\map f 1 = y$
Pages in category "Path-Connected Sets"
The following 5 pages are in this category, out of 5 total.