Pointed space

In mathematics, a pointed space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x₀, that remains unchanged during subsequent discussion, and is kept track of during all operations.

Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f between a pointed space X with basepoint x₀ and a pointed space Y with basepoint y₀ is a based map if it is continuous with respect to the topologies of X and Y and if f(x₀) = y₀. This is usually denoted

Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint.

The pointed set concept is less important; it is anyway the case of a pointed discrete space.

Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point. Thus, much of homotopy theory is usually developed on pointed spaces, and then moved to relative topologies in algebraic topology.

The class of all pointed spaces forms a category Top_• with basepoint preserving continuous maps as morphisms. Another way to think about this category is as the comma category, ({•} ↓ Top) where {•} is any one point space and Top is the category of topological spaces. (This is also called a coslice category denoted {•}/Top.) Objects in this category are continuous maps {•} → X. Such morphisms can be thought of as picking out a basepoint in X. Morphisms in ({•} ↓ Top) are morphisms in Top for which the following diagram commutes:

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Wikipedia