# Two sides lemma

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## Statement

The inclusion of two adjacent sides in the unit square is equivalent to the inclusion of one side in the unit square, viz., in the inclusion of $I$ in $I \times I$.

An explicit isotopy can also be written down, which may be more convenient in some situations.

## Applications

• The inclusion of a point in the unit interval is a cofibration. To prove this, we note that this boils down to proving that two adjacent sides of the square are a retract of the unit square, which, by the two sides lemma, is equivalent to requiring that $I$ is a retract of $I \times I$ which is clearly true.

The retraction from $I \times I$ to two sides can also be written down explicitly; this is more useful at times.