# Contractibility is retract-hereditary

From Topospaces

This article gives the statement, and possibly proof, of a topological space property (i.e., contractible space) satisfying a topological space metaproperty (i.e., retract-hereditary property of topological spaces)

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## Contents

## Statement

### Property-theoretic statement

The property of topological spaces of being contractible is a retract-hereditary property of topological spaces.

### Verbal statement

Any retract of a contractible space is contractible.

## Definitions used

### Contractible space

`Further information: contractible space`

### Retract

`Further information: Retract`

### Subspace topology

`Further information: Subspace topology`

## Proof

### Proof outline

- Consider a contracting homotopy for the whole space
- Compose this with the retraction, and show that the composite is a contracting homotopy for the retract