# What is Topology of compact-convergence?

Munkres - Topology p. 283

Definition

Let $$(Y,d)$$ be a metric space and $$X$$ be a topological space. Define $$B_C(f,\epsilon)$$ as the set $$\{g\in Y^X : \sup\limits_{x\in C} \operatorname{d}(f(x),g(x)) < \epsilon \}$$ for a given compact subspace $$C$$ and $$\epsilon >0$$ and $$f\in Y^X$$.

Then, the topology generated by all $$B_C(f,\epsilon)$$ is called the "Topology of compact convergence".

How does this is a well-defined definition?

Munkres mentioned in his book that we need some topology on $$Y^X$$ making $$C(X,Y)$$ closed which is stronger than the product topology. Then, he defined 'the topology of compact convergence' as given above.

Since he considers a topology on $$Y^X$$, he didn't assume functions to be continuous, hence not bounded.

Well, if functions are not continuous, then compactness of $$C$$ no more gurantees that $$\sup_{x\in C} d(f(x),g(x))$$ exists even when $$C$$ is nonempty, and of course it does not exist when it is empty.

Is he taking the supremum over the extended real?

Or, should i take $$d$$ as a bounded metric?

What would be the definition of this that makes sense?

Off the topic, i feel like munkres define topologies that nobody uses but really useful. An example is the uniform metric. And i think 'topology of compact convergence' would be the one too. There's no definition for this topology in wikipedia..

• I think you have omitted part of the definition of the set $B_C (f,\epsilon)$. (Probably »$\sup < \epsilon$«.) Nov 10, 2013 at 10:26
• This topology is usually called topology of "uniform convergence on compacts" and is used, to be best of my knowledge, a lot, but only in the context of continuous functions. Nov 10, 2013 at 10:26

The definition given by Munkres is correct. The set $B_C(f,\epsilon)$ contains the functions $g:X\to Y$ for which $\sup_{x\in C}d\big(f(x),g(x)\big)$ exists and is less than $\epsilon$. If $g:X\to Y$ is such that the supremum doesn’t exist (or if you prefer, is infinite), then $g\notin B_C(f,\epsilon)$, that’s all.
The topology of compact covergence is defined in Wikipedia; the definition is given in terms of which sequences of functions converge rather than directly in terms of the topology, but if you compare it with Theorem $46.2$ in Munkres, you’ll see that it’s the same topology.
• @Jonathan: It’s not hard to check that the family of sets $B_C(f,\epsilon)$, where $f$ ranges over all functions from $X$ to $Y$, $C$ over all compact subsets of $X$, and $\epsilon$ over all positive reals, is a base for a topology on ${}^XY$. The topology is not in general metrizable, but in most settings of interest it is determined by the convergent sequences, though I don’t offhand remember exactly what conditions are required to ensure that. Nov 10, 2013 at 10:55