# 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 4 subcategories, out of 4 total.

### A

### E

### P

## Pages in category "Path-Connected Spaces"

The following 35 pages are in this category, out of 35 total.

### C

### E

### I

### P

- Particular Point Space is Path-Connected
- Path Components are Open iff Union of Open Path-Connected Sets
- Path Components are Open iff Union of Open Path-Connected Sets/Lemma 1
- Path Components are Open iff Union of Open Path-Connected Sets/Path Components are Open implies Space is Union of Open Path-Connected Sets
- Path Components are Open iff Union of Open Path-Connected Sets/Space is Union of Open Path-Connected Sets implies Path Components are Open
- Path-Connected iff Path-Connected to Point
- Path-Connected Set in Subspace
- Path-Connected Space is Connected
- Path-Connected Space is not necessarily Locally Path-Connected
- Path-Connectedness is Equivalence Relation
- 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