Editing Compact-open topology (section)

{{Short description|Type of topology}}
In [[mathematics]], the '''compact-open topology''' is a [[topological space|topology]] defined on the [[set (mathematics)|set]] of [[continuous function|continuous maps]] between two [[topological space]]s. The compact-open topology is one of the commonly used topologies on [[function space]]s, and is applied in [[homotopy theory]] and [[functional analysis]].  It was introduced by [[Ralph Fox]] in 1945.<ref>{{Cite journal|url=https://www.ams.org/journals/bull/1945-51-06/S0002-9904-1945-08370-0/|doi = 10.1090/S0002-9904-1945-08370-0|title = On topologies for function spaces|year = 1945|last1 = Fox|first1 = Ralph H.|journal = Bulletin of the American Mathematical Society|volume = 51|issue = 6|pages = 429–433|doi-access = free}}</ref>

If the [[codomain]] of the [[function (mathematics)|functions]] under consideration has a [[uniform space|uniform structure]] or a [[metric space|metric structure]] then the compact-open topology is the "topology of [[uniform convergence]] on [[compact set]]s." That is to say, a [[sequence]] of functions [[limit (mathematics)|converges]] in the compact-open topology precisely when it converges uniformly on every compact subset of the [[Domain of a function|domain]].<ref>{{cite book|last1=Kelley|first1=John L.|title=General topology|date=1975|publisher=Springer-Verlag|page=230}}</ref>