# The set of continuous functions between topological spaces

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## Contents

## 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 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

- Index of spaces, sets and classes

## Notes

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

## References

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