# Two sides lemma

From Topospaces

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*This article is about the statement of a simple but indispensable lemma in topology*

## 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 in .

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 is a retract of which is clearly true.

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