Difference between revisions of "C(I,X)"

Revision as of 04:47, 3 November 2016

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] be a topological space and let [ilmath]I:=[0,1]\subset\mathbb{R}[/ilmath] - the closed unit interval. Then [ilmath]C(I,X)[/ilmath] denotes the set of continuous functions between the interval, considered with the subspace topology it inherits from the reals^{[Note 1]} - as usual.

Specifically [ilmath]C(I,X)[/ilmath] or [ilmath]C([0,1],X)[/ilmath] is the space of all paths in [ilmath](X,\mathcal{ J })[/ilmath]. That is:

• if [ilmath]f:I\rightarrow X\in C(I,X)[/ilmath] then [ilmath]f[/ilmath] is a path with initial point [ilmath]f(0)[/ilmath] and final/terminal point [ilmath]f(1)[/ilmath]

It includes as a subset, [ilmath]\Omega(X,b)[/ilmath] - the set of all loops in [ilmath]X[/ilmath] based at [ilmath]b[/ilmath]^{[Note 2]} - for all [ilmath]b\in X[/ilmath].

See also

• The set of continuous functions between topological spaces
• [ilmath]\Omega(X,b)[/ilmath]
  • The fundamental group, [ilmath]\pi_1(X,b)[/ilmath], which is the quotient of [ilmath]\Omega(X,b)[/ilmath] with the equivalence relation of end point preserving homotopic loops.
• Index of spaces, sets and classes

Notes

1. ↑ That topology is that generated by the metric [ilmath]\vert\cdot\vert[/ilmath] - absolute value.
2. ↑ A loop is a path where [ilmath]f(0)=f(1)[/ilmath], the loop is said to be based at [ilmath]b:=f(0)=f(1)[/ilmath]

References