In mathematics, a pointed space is a topological space X with a distinguished basepoint x₀ in X. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f(x₀) = y₀. This is usually denoted For other meanings of mathematics or math, see mathematics (disambiguation). ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology, a continuous function is generally defined as one for which preimages of open sets are open. ...

f : (X, x₀) → (Y, y₀).

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. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... An illustration of a homotopy between the two bold paths In topology, two continuous functions from one topological space to another are called homotopic (Greek homeos = identical and topos = place) if one can be continuously deformed into the other, such a deformation being called a homotopy between the two functions. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ...

The pointed set concept is less important; it is anyway the case of a pointed discrete space. In topology and related fields of mathematics, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points are isolated from each other in a certain sense. ...

Category of pointed spaces

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: In set theory and its applications throughout mathematics, a class is a collection of sets (or sometimes other mathematical objects) that can be unambiguously defined by a property that all its members share. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes which preserve structure. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... A comma category is a construction in category theory, a branch of mathematics. ... The category Top has topological spaces as objects and continuous maps as morphisms. ... A comma category is a construction in category theory, a branch of mathematics. ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

It is easy to see that commutativity of the diagram is equivalent to the condition that f preserves basepoints.

Note that as a pointed space {•} is a zero object in Top_• while it is only a terminal object in Top. In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C... In mathematics, an initial object of a category C is an object I in C such that to every object X in C, there exists precisely one morphism I → X. The dual notion is that of a terminal object: T is terminal, if to every object X in C...

There is a forgetful functor Top_• → Top which "forgets" which point is the basepoint. This functor has a left adjoint which assigns to each topological space X the disjoint union of X and a one point space {•} whose single element is taken to be the basepoint. A forgetful functor is a type of functor in mathematics. ... The existence of many pairs of adjoint functors is a major observation of the branch of mathematics known as category theory. ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

Operations on pointed spaces

• A subspace of a pointed space X is a topological subspace A ⊆ X which shares its basepoint with X so that the inclusion map is basepoint preserving.
• One can form the quotient of a pointed space X under any equivalence relation. The basepoint of the quotient is the image of the basepoint in X under the quotient map.
• One can form the product of two pointed spaces (X, x₀), (Y, y₀) as the topological product X × Y with (x₀, y₀) serving as the basepoint.
• The coproduct in the category of pointed spaces is the wedge sum, which can be thought of as the one-point union of spaces.
• The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum.
• The reduced suspension ΣX of a pointed space X is (up to a homeomorphism) the smash product of X and the pointed circle S¹.