nLab
locally path-connected space

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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 xx and neighbourhood VxV \ni x, there exists a path-connected neighbourhood UsubsubsetVU \subsubset V that contains xx.

Properties

Lemma

Let XX be a locally path connected space. Then the path connected component P xXP_x \subset X over any point xXx \in X is an open set.

Proof

It is sufficient to show that every point yP xy \in P_x has an neighbourhood U yU_y which is still contained in P xP_x. But by local path connectedness, yy has a neighbourhood V yV_y which is path connected. It follows by concatenation of paths that V yP xV_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 UU is connected. Hence we have a proof by contradiction.

Examples

Examples

(Euclidean space is locally path-connected)

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

Similarly the open intevals?, closed intervals and half-open intervals, regarded as topological subspaces of the Euclidean real line, are locally path connected.

Example

(open subspace of locally path-connected space is locally path connected)

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

Example

(circle is locally path-connected)

The Euclidean circle

S 1={x 2|x=1} 2 S^1 = \left\{ x \in \mathbb{R}^2 \;\vert\; {\Vert x\Vert} = 1\right\} \subset \mathbb{R}^2

is locally path connected.

Proof

By definition of the subspace topology and the defining topological base of the Euclidean plane, a base for the topology of S 1S^1 is given by the images of open intervals under the local homeomorphism

(cos(),sin()): 1S 1. (cos(-), sin(-)) \;\colon\; \mathbb{R}^1 \to S^1 \,.

But these open intervals are locally path connected by example 1, and in fact they are, evidently path-connected topological space.

The condition is a necessary assumption in the

in the form of the condition for

Revised on July 3, 2017 09:30:00 by Urs Schreiber (88.77.226.246)