algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

, a branch of mathematics, the path space

PX

of a based space

(X, *)

is the space that consists of all maps

f

from the interval

I =

, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...

/math> to ''X'' such that

f(0) = *

, called paths.Martin Frankland
Math 527 - Homotopy Theory - Fiber sequences
/ref> In other words, it is the mapping space from

(I, 0)

(X, *)

. The space

X^I

of all maps from

I

to ''X'' ( free paths or just paths) is called the free path space of ''X''. The path space

PX

can then be viewed as the pullback of

X^I \to X, \, \chi \mapsto \chi(0)

along

* \hookrightarrow X

. The natural map

PX \to X, \, \chi \to \chi(1)

is a fibration called the

path space fibration In algebraic topology, the path space fibration over a based space (X, *) is a fibration of the form :\Omega X \hookrightarrow PX \overset\to X where *PX is the path space of ''X''; i.e., PX = \operatorname(I, X) = \ equipped with the compact-open ...

References

Further reading