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

Latest revision as of 05:02, 3 November 2016

Definition

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

That is to say:

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

See also

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

Notes

1. Jump up ↑ Both $V$ and $W$ must be over the same field

References