# Space in which every retraction is a deformation retraction

## Definition

A space in which every retraction is a deformation retraction is a topological space $X$ with the property that any retraction math>r:X \to Y[/itex] for a subspace $Y$ of $X$ (with the subspace topology) arises as a deformation retraction, i.e., there is a homotopy from the identity map to that retraction that restricts to $r$ on the subspace $Y$ at all time.

## Relation with other properties

### Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
topologically convex space Template:Intermediate notions

### Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions