Compact-open topology

Let {{mvar|X}} and {{mvar|Y}} be two topological spaces, and let {{math|C(X, Y)}} denote the set of all continuous maps between {{mvar|X}} and {{mvar|Y}}. Given a compact subset {{mvar|K}} of {{mvar|X}} and an open subset {{mvar|U}} of {{mvar|Y}}, let {{math|V(K, U)}} denote the set of all functions {{math| f  ∈ C(X, Y)}} such that {{math| f (K) ⊆ U.}} In other words, $V(K,\; U)\; =\; C(K,\; U)\; \backslash times\_\{C(K,\; Y)\}\; C(X,\; Y)$. Then the collection of all such {{math|V(K, U)}} is a subbase for the compact-open topology on {{math|C(X, Y)}}. (This collection does not always form a base for a topology on {{math|C(X, Y)}}.)

When working in the category of compactly generated spaces, it is common to modify this definition by restricting to the subbase formed from those {{mvar|K}} that are the image of a compact Hausdorff space. Of course, if {{mvar|X}} is compactly generated and Hausdorff, this definition coincides with the previous one. However, the modified definition is crucial if one wants the convenient category of compactly generated weak Hausdorff spaces to be Cartesian closed, among other useful properties.{{cite journal |jstor=1995173 |title=Classifying Spaces and Infinite Symmetric Products | pages=273–298|last1=McCord |first1=M. C. |journal=Transactions of the American Mathematical Society |year=1969 |volume=146 |doi=10.1090/S0002-9947-1969-0251719-4 |doi-access=free }}{{cite web |url=http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf |title=A Concise Course in Algebraic Topology}}{{cite web |url=http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |title=Compactly Generated Spaces |access-date=2012-01-14 |archive-date=2016-03-03 |archive-url=https://web.archive.org/web/20160303174529/http://neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf |url-status=dead }} The confusion between this definition and the one above is caused by differing usage of the word compact.

If {{mvar|X}} is locally compact, then $X\; \backslash times\; -$ from the category of topological spaces always has a right adjoint $Hom(X,\; -)$. This adjoint coincides with the compact-open topology and may be used to uniquely define it. The modification of the definition for compactly generated spaces may be viewed as taking the adjoint of the product in the category of compactly generated spaces instead of the category of topological spaces, which ensures that the right adjoint always exists.

• If {{math|*}} is a one-point space then one can identify {{math|C(*, Y)}} with {{mvar|Y}}, and under this identification the compact-open topology agrees with the topology on {{mvar|Y}}. More generally, if {{mvar|X}} is a discrete space, then {{math|C(X, Y)}} can be identified with the cartesian product of {{math|{{!}}X{{!}}}} copies of {{mvar|Y}} and the compact-open topology agrees with the product topology.
• If {{mvar|Y}} is {{math|T₀}}, {{math|T₁}}, Hausdorff, regular, or Tychonoff, then the compact-open topology has the corresponding separation axiom.
• If {{mvar|X}} is Hausdorff and {{mvar|S}} is a subbase for {{mvar|Y}}, then the collection {{math|{V(K, U) : U ∈ S, K compact} }}is a subbase for the compact-open topology on {{math|C(X, Y)}}.{{cite journal |jstor=2032279 |title=Spaces of Mappings on Topological Products with Applications to Homotopy Theory |author=Jackson, James R. |journal=Proceedings of the American Mathematical Society |year=1952 |volume=3 |issue=2 |pages=327–333 |doi=10.1090/S0002-9939-1952-0047322-4 | url=https://www.ams.org/journals/proc/1952-003-02/S0002-9939-1952-0047322-4/S0002-9939-1952-0047322-4.pdf|doi-access=free }}
• If {{mvar|Y}} is a metric space (or more generally, a uniform space), then the compact-open topology is equal to the topology of compact convergence. In other words, if {{mvar|Y}} is a metric space, then a sequence {{math|{ f_n } }}converges to {{math| f }} in the compact-open topology if and only if for every compact subset {{mvar|K}} of {{mvar|X}}, {{math|{ f_n } }}converges uniformly to {{math| f }} on {{mvar|K}}. If {{mvar|X}} is compact and {{mvar|Y}} is a uniform space, then the compact-open topology is equal to the topology of uniform convergence.
• If {{math|X, Y}} and {{mvar|Z}} are topological spaces, with {{mvar|Y}} locally compact Hausdorff (or even just locally compact preregular), then the composition map {{math|C(Y, Z) × C(X, Y) → C(X, Z),}} given by {{math|( f , g) ↦  f ∘ g,}} is continuous (here all the function spaces are given the compact-open topology and {{math|C(Y, Z) × C(X, Y)}} is given the product topology).
• If {{mvar|X}} is a locally compact Hausdorff (or preregular) space, then the evaluation map {{math|e : C(X, Y) × X → Y}}, defined by {{math|e( f , x) {{=}}  f (x)}}, is continuous. This can be seen as a special case of the above where {{mvar|X}} is a one-point space.
• If {{mvar|X}} is compact, and {{mvar|Y}} is a metric space with metric {{mvar|d}}, then the compact-open topology on {{math|C(X, Y)}} is metrizable, and a metric for it is given by {{math|e( f , g) {{=}} sup{d( f (x), g(x)) : x in X},}} for {{math| f , g}} in {{math|C(X, Y)}}. More generally, if {{mvar|X}} is hemicompact, and {{mvar|Y}} metric, the compact-open topology is metrizable by the construction linked here.

The compact open topology can be used to topologize the following sets:{{Cite book|last1=Fomenko|first1=Anatoly|title=Homotopical Topology|last2=Fuchs|first2=Dmitry|edition=2nd|pages=20–23}}

• $\backslash Omega(X,x\_0)\; =\; \backslash \{\; f:\; I\; \backslash to\; X\; \backslash mid\; f(0)\; =\; f(1)\; =\; x\_0\; \backslash \}$, the loop space of $X$ at $x\_0$,
• $E(X,\; x\_0,\; x\_1)\; =\; \backslash \{\; f:\; I\; \backslash to\; X\; \backslash mid\; f(0)\; =\; x\_0\; \backslash text\{\; and\; \}\; f(1)\; =\; x\_1\; \backslash \}$,
• $E(X,\; x\_0)\; =\; \backslash \{\; f:\; I\; \backslash to\; X\; \backslash mid\; f(0)\; =\; x\_0\; \backslash \}$.

In addition, there is a homotopy equivalence between the spaces $C(\backslash Sigma\; X,\; Y)\; \backslash cong\; C(X,\; \backslash Omega\; Y)$. The topological spaces $C(X,Y)$ are useful in homotopy theory because they can be used to form a topological space and a model for the homotopy type of the set of homotopy classes of maps{{clarify|reason=This sentence is not very comprehensible. Also, the following math seems notationally confused|date=March 2025}}

$$\backslash pi(X,Y)\; =\; \backslash \{[f]:\; X\; \backslash to\; Y\; \backslash mid\; f\; \backslash text\{\; is\; a\; homotopy\; class\}\backslash \}.$$

This is because $\backslash pi(X,Y)$ is the set of path components in $C(X,Y)${{endash}}that is, there is an isomorphism of sets

$$\backslash pi(X,Y)\; \backslash to\; C(I,\; C(X,\; Y))/\{\backslash sim\},$$

where $\backslash sim$ is the homotopy equivalence.

Let {{mvar|X}} and {{mvar|Y}} be two Banach spaces defined over the same field, and let {{math|C^m(U, Y)}} denote the set of all {{mvar|m}}-continuously Fréchet-differentiable functions from the open subset {{math|U ⊆ X}} to {{mvar|Y}}. The compact-open topology is the initial topology induced by the seminorms

:$p\_\{K\}(f)\; =\; \backslash sup\; \backslash left\backslash \{\; \backslash left\backslash |\; D^j\; f(x)\; \backslash right\backslash |\; \backslash \; :\; \backslash \; x\; \backslash in\; K,\; 0\; \backslash leq\; j\; \backslash leq\; m\; \backslash right\backslash \}$

where {{math|D⁰ f (x) {{=}}  f (x)}}, for each compact subset {{math|K ⊆ U}}.{{clarification needed|date=February 2022|reason=Is this original research showing that this definition is equivalent in this special case to the general definition given above? Or is it a definition copied from an external reference, in which case that reference should be cited?}}

• Topology of uniform convergence
• {{annotated link|Uniform convergence}}

{{reflist}}

• {{Cite book|first=J.|last=Dugundji|author-link=James Dugundji|title=Topology|publisher=Allyn and Becon|year=1966|asin=B000KWE22K}}
• O.Ya. Viro, O.A. Ivanov, V.M. Kharlamov and N.Yu. Netsvetaev (2007) [http://www.math.uu.se/~oleg/topoman.html Textbook in Problems on Elementary Topology].
• {{planetmath reference|urlname=CompactOpenTopology|title=Compact-open topology}}
• [http://groupoids.org.uk/topgpds.html Topology and Groupoids Section 5.9 ] Ronald Brown, 2006

Category:General topology

Category:Topology of function spaces