Pointed space

Builds on Topological space Continuous map

Definition

A pointed space is a pair $(X, x_0)$ where $X$ is a topological space and $x_0 \in X$ . The point $x_0$ is the base point. A pointed map $f \colon (X, x_0) \to (Y, y_0)$ is a continuous map $f \colon X \to Y$ with $f(x_0) = y_0$ .

Remarks

Pointed spaces and pointed maps form a category, often written $\mathbf{Top}_*$ . A pointed homotopy between pointed maps is a homotopy relative to the base point, and $[X, Y]_*$ denotes the set of pointed homotopy classes. Example: $(S^n, e_1)$ with $e_1 = (1, 0, \ldots, 0)$ is a pointed space. Non-example: the empty space has no point, so it carries no structure of a pointed space. Every space $X$ gives a pointed space $X_+ = X \sqcup \set{*}$ with a new base point $*$ .

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Continuous map concept
Topological space concept
Pointed space definition

Tag	`Q0PAZ5QG79`
URI	`m2://Q0PAZ5QG79`
Kind	definition
Writing	W0 stub
MSC	55P99, 55Q05
Defines	Pointed space
Uses	Topological space, Continuous map
Changed	2 November 2026

Pointed space

Definition

Remarks

Reading path

Further reading

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