Category of pointed spaces | Wikipedia audio article
Channel: wikipedia tts
Date: 17/05/2019

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Pointed_space


00:01:13 Category of pointed spaces
00:02:21 Operations on pointed spaces



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SUMMARY
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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

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.
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.