Compact open topology

The compact open topology for short KO topology is a structure on function spaces of continuous functions considered in the mathematical sub-area of topology . Namely, if and are topological spaces , then the continuous mappings are the structure-preserving maps. It therefore makes sense to equip the set of all continuous functions with a topology again. Among the many ways of doing this, the compact open topology has proven to be particularly suitable. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle C(X,Y)}$ ${\displaystyle X\to Y}$

The mathematicians RH Fox (1945) and Richard Friederich Arens (1946) were the first to define this topology and to study it systematically.

definition

Be and topological spaces. Is compact and open , so be . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle K\subset X}$ ${\displaystyle U\subset Y}$ ${\displaystyle \Omega (K,U):=\{f\in C(X,Y):\,f(K)\subset U\}}$

The compact open topology on is the topology generated by all sets of form , compact, open, i.e. That is, the open sets of this topology are arbitrary unions of finite intersections of such sets . ${\displaystyle C(X,Y)}$ ${\displaystyle \Omega (K,U)}$ ${\displaystyle K\subset X}$ ${\displaystyle U\subset Y}$ ${\displaystyle \Omega (K,U)}$

The quantities , compact, open, thus form a sub-basis of the compact-open topology. This topology is often abbreviated (compact-open), then designates the space that is provided with the compact-open topology. ${\displaystyle \Omega (K,U)}$ ${\displaystyle K\subset X}$ ${\displaystyle U\subset Y}$ ${\displaystyle co}$ ${\displaystyle C_{co}(X,Y)}$ ${\displaystyle C(X,Y)}$

properties

In the following let and be topological spaces. ${\displaystyle X}$ ${\displaystyle Y}$

Separation axioms

If Y is T ₀ space , T ₁ space , Hausdorff space , regular space or a completely regular space , then the same axiom of separation suffices . ${\displaystyle C_{co}(X,Y)}$

The evaluation illustration

For each subset you have the evaluation mapping . Is in any topology such that is continuous ( carrying case the product topology from and on the given topology) is , d. i.e. the relative compact open topology on is coarser than . In an important special case, the evaluation mapping is continuous if the relative compact-open topology is provided; the following applies: ${\displaystyle H\subset C(X,Y)}$ ${\displaystyle j_{H}:H\times X\to Y,(f,x)\mapsto f(x)}$ ${\displaystyle \tau }$ ${\displaystyle H}$ ${\displaystyle j_{H}}$ ${\displaystyle H\times X}$ ${\displaystyle \tau }$ ${\displaystyle X}$ ${\displaystyle co|_{H}\subset \tau }$ ${\displaystyle H}$ ${\displaystyle \tau }$ ${\displaystyle j_{H}}$ ${\displaystyle H}$

If locally compact and any topological space, then the compact open topology is the coarsest topology on each subset , which makes the evaluation mapping continuous. ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle H\subset C(X,Y)}$ ${\displaystyle j_{H}:H\times X\to Y,(f,x)\mapsto f(x)}$

composition

Be and locally compact, be a third topological space. Then is the composition picture ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$

${\displaystyle C_{co}(X,Y)\times C_{co}(Y,Z)\rightarrow C_{co}(X,Z),\,\,(f,g)\mapsto g\circ f}$

steadily.

Compact convergence

Be locally compact, uniform space . Then the compact open topology coincides with the topology of compact convergence . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle C(X,Y)}$

application

The recursive definition of the higher homotopy groups is presented here as a typical application in algebraic topology . Let it be a topological space with an excellent point . With the going fundamental group to the base point indicated. For the definition of the higher homotopy groups , consider the space of all continuous maps of the unit square to which the edge of the unit square on the base point represent. If one denotes the constant function , which maps the unit square onto the point , and if one provides the relative compact-open topology of , then the pair is a topological space with a marked point. ${\displaystyle X}$ ${\displaystyle p\in X}$ ${\displaystyle \pi _{1}(X,p)}$ ${\displaystyle p}$ ${\displaystyle \pi _{n}(X,p)}$ ${\displaystyle \Omega _{X,p}}$ ${\displaystyle g:([0,1]^{2},\partial [0,1]^{2})\to (X,p)}$ ${\displaystyle [0,1]^{2}}$ ${\displaystyle X}$ ${\displaystyle \partial [0,1]^{2}}$ ${\displaystyle p}$ ${\displaystyle \Omega _{X,p}}$ ${\displaystyle p}$ ${\displaystyle {\tilde {p}}}$ ${\displaystyle \Omega _{X,p}}$ ${\displaystyle C([0,1]^{2},X)}$ ${\displaystyle (\Omega _{X,p},{\tilde {p}})}$

One defines now and more generally recursively for . ${\displaystyle \pi _{2}(X,p):=\pi _{1}(\Omega _{X,p},{\tilde {p}})}$ ${\displaystyle \pi _{n}(X,p):=\pi _{n-1}(\Omega _{X,p},{\tilde {p}})}$ ${\displaystyle n>1}$

literature

• Johann Cigler , Hans-Christian Reichel : Topology. A basic lecture (=  BI university paperbacks . Volume 121 ). Bibliographisches Institut, Mannheim u. a. 1978, ISBN 3-411-00121-6 .
• Horst Schubert : Topology. An introduction (= Teubner's mathematical guidelines. ZDB -ID 259127-3 ). Teubner, Stuttgart 1964, (4th edition. Teubner, Stuttgart 1975, ISBN 3-519-12200-6 ).

Individual evidence

1. ↑ Gerd Laures, Markus Szymik: Basic Course topology . Spectrum - Akademischer Verlag, Heidelberg 2009, ISBN 978-3-8274-2040-4 , p. 72 .
2. ^ Boto von Querenburg : Set theoretical topology. 3rd, revised and expanded edition. Springer, Berlin a. a. 2001, ISBN 3-540-67790-9 , p. 333.