This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Pointed space" – news ·newspapers ·books ·scholar ·JSTOR(November 2009) (Learn how and when to remove this message)

Inmathematics, apointed space orbased space is atopological space with a distinguished point, thebasepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as${\displaystyle x_0,}$ that remains unchanged during subsequent discussion, and is kept track of during all operations.

Maps of pointed spaces (based maps) arecontinuous maps preserving basepoints, i.e., a map${\displaystyle f}$ between a pointed space${\displaystyle X}$ with basepoint${\displaystyle x_0}$ and a pointed space${\displaystyle Y}$ with basepoint${\displaystyle y_0}$ is a based map if it is continuous with respect to the topologies of${\displaystyle X}$ and${\displaystyle Y}$ and if${\displaystyle f\left(x_0\right)=y_0.}$ This is usually denoted

${\displaystyle f:\left(X,x_0\right)\to \left(Y,y_0\right).}$

Pointed spaces are important inalgebraic topology, particularly inhomotopy theory, where many constructions, such as thefundamental group, depend on a choice of basepoint.

Thepointed set concept is less important; it is anyway the case of a pointeddiscrete space.

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

Theclass of all pointed spaces forms acategoryTop_{${\displaystyle \bullet }$} with basepoint preserving continuous maps asmorphisms. Another way to think about this category is as thecomma category, (${\displaystyle \{\bullet \}\downarrow }$Top) where${\displaystyle \{\bullet \}}$ is any one point space andTop is thecategory of topological spaces. (This is also called acoslice category denoted${\displaystyle \{\bullet \}/}$Top.) Objects in this category are continuous maps${\displaystyle \{\bullet \}\to X.}$ Such maps can be thought of as picking out a basepoint in${\displaystyle X.}$ Morphisms in (${\displaystyle \{\bullet \}\downarrow }$Top) are morphisms inTop for which the following diagramcommutes:

It is easy to see that commutativity of the diagram is equivalent to the condition that${\displaystyle f}$ preserves basepoints.

As a pointed space,${\displaystyle \{\bullet \}}$ is azero object inTop_{${\displaystyle \{\bullet \}}$}, while it is only aterminal object inTop.

There is aforgetful functorTop_{${\displaystyle \{\bullet \}}$}${\displaystyle \to }$Top which "forgets" which point is the basepoint. This functor has aleft adjoint which assigns to each topological space${\displaystyle X}$ thedisjoint union of${\displaystyle X}$ and a one-point space${\displaystyle \{\bullet \}}$ whose single element is taken to be the basepoint.

• Asubspace of a pointed space${\displaystyle X}$ is atopological subspace${\displaystyle A\subseteq X}$ which shares its basepoint with${\displaystyle X}$ so that theinclusion map is basepoint preserving.
• One can form thequotient of a pointed space${\displaystyle X}$ under anyequivalence relation. The basepoint of the quotient is the image of the basepoint in${\displaystyle X}$ under the quotient map.
• One can form theproduct of two pointed spaces${\displaystyle \left(X,x_0\right),}$${\displaystyle \left(Y,y_0\right)}$ as thetopological product${\displaystyle X\times Y}$ with${\displaystyle \left(x_0,y_0\right)}$serving as the basepoint.
• Thecoproduct in the category of pointed spaces is thewedge sum, which can be thought of as the 'one-point union' of spaces.
• Thesmash product of two pointed spaces is essentially thequotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into asymmetric monoidal category with the pointed0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail. But it is true for some more restricted categories of spaces, such ascompactly generatedweak Hausdorff ones.
• Thereduced suspension${\displaystyle \mathrm{\Sigma }X}$ of a pointed space${\displaystyle X}$ is (up to ahomeomorphism) the smash product of${\displaystyle X}$ and the pointed circle${\displaystyle S^1.}$
• The reduced suspension is a functor from the category of pointed spaces to itself. This functor isleft adjoint to the functor${\displaystyle \mathrm{\Omega }}$ taking a pointed space${\displaystyle X}$ to itsloop space${\displaystyle \mathrm{\Omega }X}$.

• Category of groups – category of groups and group homomorphismsPages displaying wikidata descriptions as a fallback
• Category of metric spaces – mathematical category with metric spaces as its objects and distance-non-increasing maps as its morphismsPages displaying wikidata descriptions as a fallback
• Category of sets – Category in mathematics where the objects are sets
• Category of topological spaces – category whose objects are topological spaces and whose morphisms are continuous mapsPages displaying wikidata descriptions as a fallback
• Category of topological vector spaces – Topological category

• Gamelin, Theodore W.; Greene, Robert Everist (1999) [1983].Introduction to Topology (second ed.).Dover Publications.ISBN 0-486-40680-6.
• Mac Lane, Saunders (September 1998).Categories for the Working Mathematician (second ed.). Springer.ISBN 0-387-98403-8.

• mathoverflow discussion on several base points and groupoids

• Topology
• Homotopy theory
• Categories in category theory
• Topological spaces

• Articles with short description
• Short description matches Wikidata
• Articles needing additional references from November 2009
• All articles needing additional references
• Pages displaying wikidata descriptions as a fallback via Module:Annotated link