# Contents

## Definition

A topological space is called locally path-connected if it has a basis of path-connected neighbourhoods. In other words, if for every point $x$ and neighbourhood $V \ni x$, there exists a path-connected neighbourhood $U \subset V$ that contains $x$.

## Properties

###### Lemma

Let $X$ be a locally path connected space. Then the path connected component $P_x \subset X$ over any point $x \in X$ is an open set.

###### Proof

It is sufficient to show that every point $y \in P_x$ has an neighbourhood $U_y$ which is still contained in $P_x$. But by local path connectedness, $y$ has a neighbourhood $V_y$ which is path connected. It follows by concatenation of paths that $V_y \subset P_x$.

A locally path-connected space is connected if and only if it is path-connected.

###### Proposition

The connected components of a locally path-connected space are the same as its path-connected components.

###### Proof

A path connected component is always connected (this lemma), and in a locally path-connected space is it also open (lemma 1). This means that every path-connected component is also connected.

Conversely, it is now sufficient to see that every connected component is path-connected. Suppose it were not, then it would be covered by more than one disjoint non-empty path-connected components. But by lemma 1 these would be all open. This would be in contradiction with the assumption that $U$ is connected. Hence we have a proof by contradiction.

## Examples

###### Examples

(Euclidean space is locally path-connected)

For $n \in \mathbb{N}$ then Euclidean space $\mathbb{R}^n$ (with its metric topology) is locally path-connected, since each open ball is path-connected topological space.

###### Example

Every open subspace of a locally path connected topological space is itself locally path connected.

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Revised on June 13, 2017 04:11:10 by Urs Schreiber (46.183.103.17)