# Category:Definitions/Path-Connected Spaces

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This category contains definitions related to Path-Connected Spaces.

Related results can be found in Category: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 only the following subcategory.

### A

## Pages in category "Definitions/Path-Connected Spaces"

The following 25 pages are in this category, out of 25 total.

### L

### P

- Definition:Path (Topology)
- Definition:Path (Topology)/Endpoint
- Definition:Path (Topology)/Final Point
- Definition:Path (Topology)/Initial Point
- Definition:Path Component
- Definition:Path Component (Topology)
- Definition:Path Component/Equivalence Class
- Definition:Path Component/Maximal Path-Connected Set
- Definition:Path Component/Union of Path-Connected Sets
- Definition:Path Components (Topology)
- Definition:Path-Connected
- Definition:Path-Connected Points
- Definition:Path-Connected Set (Topology)
- Definition:Path-Connected Space
- Definition:Path-Connected/Metric Space
- Definition:Path-Connected/Metric Space/Subset
- Definition:Path-Connected/Topology
- Definition:Path-Connected/Topology/Points
- Definition:Path-Connected/Topology/Set
- Definition:Path-Connected/Topology/Topological Space