Difference between revisions of "The set of continuous functions between topological spaces"

Revision as of 05:01, 3 November 2016

Definition

Let [ilmath](X,\mathcal{ J })[/ilmath] and [ilmath](Y,\mathcal{ K })[/ilmath] be topological spaces. Then [ilmath]C(X,Y)[/ilmath] denotes the set of all continuous functions from [ilmath]X[/ilmath] to [ilmath]Y[/ilmath], with respect to the topologies: [ilmath]\mathcal{J} [/ilmath] and [ilmath]\mathcal{K} [/ilmath].

That is to say:

• [ilmath]\big(f\in C(X,Y)\big)\iff\big(f:X\rightarrow Y\text{ is a continuous function}\big)[/ilmath]

See also

• Subsets:
  • [ilmath]C([0,1],X)[/ilmath] - all paths in [ilmath]X[/ilmath]
  • [ilmath]\Omega(X,b)[/ilmath] - all loops in [ilmath]X[/ilmath] based at [ilmath]b\in X[/ilmath]
    • The fundamental group
• The space of all continuous linear maps - denoted [ilmath]\mathcal{L}(V,W)[/ilmath] for vector spaces [ilmath]V[/ilmath] and [ilmath]W[/ilmath] usually over either the field of the reals or complex numbers^{[Note 1]}
  • The space of all linear maps - denote [ilmath]L(V,W)[/ilmath], for [ilmath]V[/ilmath] and [ilmath]W[/ilmath] vector spaces over the same field

Notes

1. ↑ Both [ilmath]V[/ilmath] and [ilmath]W[/ilmath] must be over the same field

References