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# homotopy extension property

Let $X$ be a topological space and $A$ a subspace of $X$. Suppose there is a continuous map $f:X\to Y$ and a homotopy of maps $F:A\times I\to Y$. The inclusion map $i:A\to X$ is said to have the *homotopy extension property* if there exists a continuous map $F^{{{}^{{\prime}}}}$ such that the following diagram commutes:

$\SelectTips{eu}{}\xymatrix{&A\ar[d]^{{i}}\ar[r]^{{i_{{0}}}}&A\times I\ar[d]^{{% F}}\ar@/^{1}.5pc/[dd]^{{i\times id_{{I}}}}\\ &X\ar[r]^{{f}}\ar[dr]^{{i_{{0}}}}&Y\\ &&X\times I\ar[u]_{{F^{{{}^{{\prime}}}}}}}$ |

Here, $i_{0}=(x,0)$ for all $x\in X$.

Related:

Cofibration, HomotopyLiftingProperty

Type of Math Object:

Definition

Major Section:

Reference

## Mathematics Subject Classification

55P05*no label found*

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new correction: examples and OEIS sequences by fizzie

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new correction: Define Galois correspondence by porton

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new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

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new correction: Many corrections by Smarandache

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new question: how to contest an entry? by zorba

new question: simple question by parag