# Category:Path Components

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This category contains results about Path Components in the context of Topology.

Let $\sim$ be the equivalence relation on $T$ defined as:

- $x \sim y \iff x$ and $y$ are path-connected.

The equivalence classes of $\sim$ are called the **path components of $T$**.

If $x \in T$, then the **path component of $T$** containing $x$ (that is, the set of points $y \in T$ with $x \sim y$) can be denoted by $\operatorname{PC}_x \left({T}\right)$.

## Pages in category "Path Components"

The following 9 pages are in this category, out of 9 total.

### P

- Path Component is not necessarily Arc Component
- Path Component of Locally Path-Connected Space is Open
- 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