compact-open topology


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compact-open topology

[¦käm‚pakt ¦ō·pən tə′päl·ə·jē]
(mathematics)
A topology on the space of all continuous functions from one topological space into another; a subbase for this topology is given by the sets W (K,U) = {ƒ:ƒ(K)⊂ U }, where K is compact and U is open.
References in periodicals archive ?
In [1], some results about Segal algebras were obtained, where one of the conditions that had to be fulfilled was that the set [C.sub.0](X,K)[cross product]B (the full definition of this set will be given further in this paper) had to be dense in [C.sub.0](X,B) (in the compact-open topology) for a topological algebra B.
Consider the algebra ([C.sub.0](X,Y),[c.sub.Y]) of all continuous maps f : X [right arrow] Y vanishing at infinity equippped with the compact-open topology [c.sub.Y], where the subbase of the topology [c.sub.Y] on [C.sub.0](X, Y) consists of all sets of the form
The space C(X) endowed with the pointwise topology or with the compact-open topology is denoted by [C.sub.p](X) and [C.sub.k](X), respectively.
The space B(H) can be endowed with several topologies, among them we have the strong operator topology and the compact-open topology. These topologies are of interest when studying principal bundles and therefore are the ones of interest in this paper.
Note that in the compact-open topology a sequence of bounded operators {[T.sub.n]} converges to T [member of] B(H) if and only if [T.sub.n] [|.sub.K] [right arrow] T[|.sub.K] uniformly for every compact set K [subset] H.
Consider the algebra [C.sub.c](X) of continuous complex valued functions on X, endowed with the compact-open topology. Then [C.sub.c](X) is a unital commutative complete locally uniformly A-convex algebra.
O'Meara, On paracompactness in function spaces with the compact-open topology, Proc.
When C(X) is equipped with the compact-open topology [[tau].sub.c] we write [C.sub.c](X).
Let's recall the definitions: By O (X) we denote the algebra of holomorphic functions on X endowed with compact-open topology. By Stein's original definition, simplified by later developments, a complex manifold X is Stein (or, as Karl Stein put it, holomorphically complete) if it satisfies the following two conditions.
In this paper we consider the ring R = [SL.sub.n](O(X)) of holomorphic maps from a Stein space X to [SL.sub.n](C) endowed with compact-open topology, the natural topology for holomorphic mappings.
For a metric space, Iso(X) denotes the group of all surjective self-isometries of X equipped with the topology of pointwise convergence, induced by the embedding Iso(X) [??] [X.sup.X] (or, which is the same, compact-open topology).
introduced a topology on higher homotopy groups of a pointed space (X, x) as a quotient of Hom(([I.sup.n], [I.sup.n]), (X, x)) equipped with the compact-open topology and denoted it by [[pi].sup.top.sub.n] (X, x).