View source for C(I,X)

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==Definition==
Let {{Top.|X|J}} be a [[topological space]] and let {{M|1=I:=[0,1]\subset\mathbb{R} }} - the [[closed unit interval]]. Then {{M|C(I,X)}} denotes the [[set of continuous functions]] between the interval, considered with the [[subspace topology]] it inherits from [[the reals]]<ref group="Note">That topology is that generated by the metric {{M|\vert\cdot\vert}} - [[absolute value]].</ref> - as usual.


Specifically {{M|C(I,X)}} or {{M|C([0,1],X)}} is the space of all {{plural|path|s}} in {{Top.|X|J}}. That is:
* if {{M|f:I\rightarrow X\in C(I,X)}} then {{M|f}} is a [[path]] with initial point {{M|f(0)}} and final/terminal point {{M|f(1)}}
It includes as a [[subset of|subset]], [[Omega(X,b)|{{M|\Omega(X,b)}}]] - the set of all {{plural|loop|s}} in {{M|X}} based at {{M|b}}<ref group="Note">A loop is a path where {{M|1=f(0)=f(1)}}, the loop is said to be based at {{M|1=b:=f(0)=f(1)}}</ref> - for all {{M|b\in X}}.
==See also==
* [[The set of continuous functions between topological spaces]]
* [[Omega(X,b)|{{M|\Omega(X,b)}}]]
** [[The fundamental group]], {{M|\pi_1(X,b)}}, which is the {{link|quotient|equivalence relation}} of {{M|\Omega(X,b)}} with the [[equivalence relation]] of [[end point preserving homotopic]] loops.
* [[Index of spaces, sets and classes]]

==Notes==
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==References==
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{{Definition|Topology|Functional Analysis|Analysis}}