Difference between revisions of "C(I,X)"
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* [[Omega(X,b)|{{M|\Omega(X,b)}}]] | * [[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. | ** [[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 and classes]] | + | * [[Index of spaces, sets and classes]] |
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==Notes== | ==Notes== | ||
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Latest revision as of 05:08, 3 November 2016
Grade: B
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Something would be good, should be abundant
Contents
[hide]Definition
Let (X,J) be a topological space and let I:=[0,1]⊂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,J). That is:
- if f:I→X∈C(I,X) then f is a path with initial point f(0) and final/terminal point f(1)
It includes as a subset, Ω(X,b) - the set of all loops in X based at b[Note 2] - for all b∈X.
See also
- The set of continuous functions between topological spaces
- Ω(X,b)
- The fundamental group, π1(X,b), which is the quotient of Ω(X,b) with the equivalence relation of end point preserving homotopic loops.
- Index of spaces, sets and classes
Notes
- Jump up ↑ That topology is that generated by the metric |⋅| - absolute value.
- 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)
