Actually the motivation for studying the compact open. Basically it is given by declaring which subsets are open sets. A subspace aof x is compact if and only if every open cover of aby open sets in xhas a nite subcover. It is important to note that these sets are not closed under intersection, and do not form a topological basis.
The compactopen topology on the set of continuous functions x y is generated by the subbasis of subsets uk. A sequence of functions converges to iff for every containing contains all but a finite number of the. We begin by introducing this topology in a more general context. U, where is compact and u is an open neighborhood of 0 in t. There are ten questions, each worth ten points, so you should pace yourself at around 1012 minutes per question, since they. Co nite topology we declare that a subset u of r is open i either u.
We give a short proof that the space of continuous functions from t0. Let kbe a compact hausdor space, and let xbe an arbitrary space. While compact may infer small size, this is not true in general. The topology of convergence for this situation is called the compact open topology. The inverse image under fof every open set in yis an open set in x. We also have the following simple lemma lemma 3 a subset u of a metric space is open if and only if it is a neighbor hood of each of its points. Pdf this paper studies the ccompactopen topology on the set cx of all realvalued continuous functions on a tychonov space x and. Free topology books download ebooks online textbooks. In particular, its sufficient to make the evaluation map mathx \times \mathrmmapx,y \to. Recall that a continuum is a compact connected space. Ybe the space of continuous functions from x to y, and give this space the. Unfortunately i dont know any references for this facts, except the lecture notes of the 198283 lectures of prof. Topological spaces, product and quotient spaces, connected topological spaces, compact topological spaces, countability and separation axioms. Closed sets, hausdor spaces, and closure of a set 9 8.
It was introduced by ralph fox in 1945 if the codomain of the functions under consideration has a uniform structure or a. Spaces of mappings on topological products with applications to homotopy theory pdf. A topology on a set x is a collection tof subsets of x such that t1. Xtogether with the collection of all its open subsets a \ topological space. An elementary counterexample in the compact open topology jonathan groves abstract. For some purposes, the homeomorphism group happens to be too big, but by means of the isotopy relation, one can reduce this group to the mapping class group. In addition to metrizability, separability and second countability of this topology on kc x, various kinds of topological properties of this topology are studied in detail.
As you can see, this approach to the study of shapes involves not just elements and functions, like the theory of metric spaces, but also subsets and even collections of subsets. In this respect the compactopen topology is preferable to the topology of pointwise convergence, since the latter is. The simplest function space to compare topologies is the space of realvalued continuous functions. In general it is stronger than the compact open topology. This paper studies the compactopen topology on the set kcx of all realvalued functions defined on a tychonoff space, which are continuous on compact. A compact subset of a hausdorff space for example a metric space is always closed, so your set would be closed and open. Check our section of free ebooks and guides on topology now. Now that can happen, but in a connected space the only open and closed subsets are the empty set and the space itself.
Xcarries a natural topology, called the compactopen topology. The c o mpact open topology is one of the common ly used to pologies on function spaces, and is applied in homotopy theory and functional analysis. Complex analysisthe compactopen topology wikibooks. Part ii is an introduction to algebraic topology, which associates algebraic structures such as groups to topological spaces. The compact open topology made its appearance in 1945 in a paper by ralph h. The compact open topology is one of the commonly used topologies on function spaces, and is applied in homotopy theory and functional analysis. Ok so i want to understand why that family forms a subbasis for the compact open topology. It is cartesian closed and the topology for function spaces is the compact open topology. The claim that t care approximating is is easy to check as follows.
R with the usual topology is a compact topological space. An elementary counterexample in the compactopen topology. If x62 s c, then cdoes not cover v, hence o v is an open alexandro open containing v so v. In mathematics, the compactopen topology is a topology defined on the set of continuous. Lecture notes on topology for mat35004500 following j. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. In mathematics, the compact open topolo gy is a topol ogy defined on the set of continuous maps betw een two topolo gical s paces.
Introduction when we consider properties of a reasonable function, probably the. Function spaces in standard terminology, map mapping continuous function. On homeomorphism groups and the compactopen topology. This group can be given a topology, such as the compact open topology, which under certain assumptions makes it a topological group. Again, the compact open topology guarantees that there are enough compact sets such that the topology is actually compactly generated i guess that this is also a consequence of a version of. A set x with a topology tis called a topological space. This paper studies the compactopen topology on the set kc x of all realvalued functions defined on a tychonoff space, which are continuous on compact subsets of x.
To make use of this, we need to recognize sets of open subsets that serve as the basis for some topology. Locally compact spaces and the alexandro compacti cation58. The c compactopen topology on function spaces article pdf available in topology and its applications 159 january 2012 with 234 reads how we measure reads. A subset uof a metric space xis closed if the complement xnuis open. So, to start, the compactopen topology is the minimal as in, fewest open sets. If x is a discrete space with m points and y has n points, there are nm maps. In mathematics, the compact open topology is a topology defined on the set of continuous maps between two topological spaces. Introduction to topology tomoo matsumura november 30, 2010 contents. All articles published after 48 months have unrestricted access and will remain permanently free to read and download. Let k be a compact hausdorff space, and let x be an arbitrary space. The collection of all open subsets will be called the topology on x, and is usually denoted t.
To see that a topology is strong if and only if evaluation is continuous, notice. The compact open topology is a special case of this construction. This page contains list of freely available ebooks, online textbooks and tutorials in topology. This paper studies the compactopen topology on the set kcx of all realvalued functions defined on a tychonoff space, which are continuous. Compact open topology and evaluation map via neutrosophic sets. Topology and its applications open archive elsevier. Suppose x and y are topological spaces and cx,y is the set of continuous maps from f. Introduction to topology answers to the test questions stefan kohl. Chapter 5 compactness compactness is the generalization to topological spaces of the property of closed and bounded subsets of the real line. Open problems in topology edited by jan van mill free university amsterdam, the netherlands george m. However, in the category of compactly generated spaces it works. As you can see, this approach to the study of shapes involves not just elements and functions, like the theory of metric spaces, but also subsets and even collections of. The importance of compact open topologies is due to the fact that they are essential elements in pontryagins theory of duality of locally compact commutative groups and participate in the construction of skew products. Given spaces x and y, denote by yx the set of all maps x.
Mathematics 490 introduction to topology winter 2007 1. Exam i solutions topology math 5863 1a if xand y are topological spaces, let cx. Weak topologies, which we will investigate later are always locally convex. Topology and its applications open archive list of the recent articles made freely available as part of this journals open archive. We now write this in a fashion so that everyone, even those who are not familiar with uniform spaces, will be able. That is, the open sets in the compactopen topology are the arbitrary unions of finite intersections of. For every open set oin the topological space y,c,thesetf. Help understand compact open topology mathematics stack. Edmund hall oxford university oxford, united kingdom. U x y is open in the product topology if and only if, given any point x. Bounded set, compactopen topology, countably compact, convex strong mackey. We will prove that the only topological vector spaces that are locally compact are. A point z is a limit point for a set a if every open set u containing z.
What is the intuition behind the compactopen topology. Thus the axioms are the abstraction of the properties that open sets have. You should expect this since even though we have not mentioned it, you should expect that compactness is a topological. By a neighbourhood of a point, we mean an open set containing that point. Instead, the sets form a subbasis for the compactopen topology. Holdgruen on harmonic analysis in a shelf in the library of the mathematics institute in goettingen, because i learned this in my undergrad courses and not from books.