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Continuous Inclusion Mapping Commutative Diagram Subspace Topology Topological Space Homotopy Of Maps

1	Commutative Diagram
1	Subspace Topology
1	Topological Space
1	Continuous
1	Inclusion Mapping
4	Homotopy Of Maps

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Homotopy Extension Property

Let be a topological space and a subspace of . 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 $F^{'}$ such that the following diagram commutes:

$\displaystyle \SelectTips{eu}{} \xymatrix{ & A \ar[d]^{i} \ar[r]^{i_{0}} & A \t... ...I}} \\ & X \ar[r]^{f} \ar[dr]^{i_{0}} & Y \\ & & X \times I \ar[u]_{F^{'}} }$

Here,

for all $x \in X$ .