Compact-Open topology for the space of all continuous paths

By | October 25, 2019

As i mentioned in Topological Complexity, M.Farber shows the space of all continuous paths in space X by PX and equip the path space  PX  with compact-open topology. But why?

In this note i try to account compact-open topology and another one called compact convergence topology and obviously their relations.

Compact-Open topology

When we have a space of maps, more precisely in this case, a space of continuous functions (paths), and we want to exert a topology on this space, it is natural to considering compact-open topology. But what is it exactly?

Here i rewrite the definitiion of compact-open topology from “topology, A first course by Munkres”:

Let X and Y be topological spaces. If C is a compact subset of X and U is an open subset of Y , define

S(C,U) =\{f \mid f \in \mathcal C(X,Y) \thinspace and \thinspace f(C)\subset U \}

The sets S(C,U) form a subbasis for a topology on \mathcal C(X,Y) called the compact-open topology.

Compact convergence topology

Again from “topology, A first course by Munkres” we have:

Let (Y,d) be a metric space; let X be a topological space. Given an element f of Y^X , a compact subset C in X , and a number \epsilon > 0 , let B_{c}(f,e) denote the set of all those elements g of Y^X for which

lub\{d(f(x),g(x)) \mid x \in C \} < \epsilon

The sets B_{c}(f,e) form a basis for a topology on Y^X . It is called the topology of compact convergence (or sometimes the “topology of uniform convergence on compact sets”).

Coincidence of Compact convergence and Compact-Open topology

Theorem 5.1 in the book noted above is as below:

Let X be a space and let (Y,d) be a metric space. For the space \mathcal C(X,Y) , the compact-open topology and topology of compact convergence coincide.

With this theorem we can understand the continuity of functions showed by s_{i} in the notion of Topological Complexity, better and better. In fact with compact convergence topology on X^I , we can conceive closing of two paths more easier, because we can assume a metric on X .

But on the other hand we know that this metric is not important, means the structured topology doesn’t depend on the metric of X .

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