# Homotopy extension property

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In mathematics, 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.

## Visualisation

The homotopy extension property is depicted in the following diagram

If the above diagram (without the dashed map) commutes, which is equivalent to the conditions above, then there exists a map ${\tilde {f}}$ which makes the diagram commute. By currying, note that a map ${\tilde {f}}\colon X\to Y^{I}$ is the same as a map ${\tilde {f}}\colon X\times I\to Y$ .

Also compare this to the visualization of the homotopy lifting property.

## Other

In fact, if you consider any cofibration $i:Y\to Z$ , then we have that $\mathbf {\mathit {Y}}$ is homeomorphic to its image under $\mathbf {\mathit {i}}$ . This implies that any cofibration can be treated as an inclusion map, and therefore it can be treated as having the homotopy extension property.