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

Revision as of 04:47, 3 November 2016

Definition

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

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

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

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

See also

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

Notes

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

References