# Questions tagged [homotopy-extension-property]

In the area of algebraic topology, the homotopy extension property indicates which homotopies defined on a subspace can be extended to a homotopy defined on a larger space.

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### When does a homotopy exist?

Let $X\coprod X$ be the disjoint union of $X$ with itself and let there be a commutative square \require{AMScd} \begin{CD} X \coprod X @>{f,g}>> A \\ @V{(i_0, i_1)}VV @VV{p}V \\ X \times ...
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### If a homotopy can be extended to a neighborhood of a closed subspace of a normal space $X$, then it can be extended to all of $X$.

During some self-study, I came across the following problem in Spanier's Algebraic Topology: Statement: Suppose $X$ is a normal space, and $A$ is a closed subspace of $X$. Let $f\colon X \to Y$ be a ...
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### any cofibration $i:A \to B$ is a homeomorphism onto its image (question regarding the inverse map)

I was recently working on a problem that introduced the homotopy extension property as a cofibration $i:A \to B$. Let's say we are given the commutative diagram: Now, if $i:A \to B$ is the inclusion ...
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### $(M_f, X \cup Y)$ has the homotopy extension property

Let $X,Y$ be spaces and $f:X \to Y$ a continuous map. I want to show that $(M_f, X \cup Y)$ has the homotopy extension property. In the proof of Whitehead's theorem (Theorem 4.5 in Hatcher's ...