is a continuous function, possibly with some extra constraints.

Often, one works with a pointed space--that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserves basepoints; that is, it sends the basepoint of the domain to that of the codomain. In contrast, a free map is one which needn't preserve basepoints.

The Cartesian product of two pointed spaces are not naturally pointed. A substitute is the smash product

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