Category:Path-Connected Spaces
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This category contains results about Path-Connected Spaces.
Definitions specific to this category can be found in Definitions/Path-Connected Spaces.
Let $T = \struct {S, \tau}$ be a topological space.
Then $T$ is a path-connected space if and only if $S$ is a path-connected set of $T$.
That is, $T$ is a path-connected space if and only if:
- for every $x, y \in S$, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that:
- $\map f 0 = x$
- and:
- $\map f 1 = y$
Subcategories
This category has the following 10 subcategories, out of 10 total.
A
C
- Connected Manifolds (1 P)
P
- Path-Connected Sets (5 P)
- Paths (Topology) (1 P)
S
- Simply Connected Spaces (11 P)
Pages in category "Path-Connected Spaces"
The following 37 pages are in this category, out of 37 total.
C
- Closed Ball is Path-Connected
- Connected and Locally Path-Connected Implies Path Connected
- Connected Open Subset of Euclidean Space is Path-Connected
- Continuous Image of Path-Connected Set is Path-Connected
- Continuous Image of Path-Connected Set is Path-Connected/Metric Space
- Convex Set is Path-Connected
- Countable Finite Complement Space is not Path-Connected
E
I
L
P
- Particular Point Space is Path-Connected
- Path Components are Open iff Union of Open Path-Connected Sets
- Path-Connected iff Path-Connected to Point
- Path-Connected Set in Subspace
- Path-Connected Space is not necessarily Locally Path-Connected
- Path-Connectedness is Equivalence Relation
- Path-Connectedness is Preserved under Homeomorphism
- Point is Path-Connected to Itself
- Points are Path-Connected iff Contained in Path-Connected Set
- Product Space is Path-connected iff Factor Spaces are Path-connected