Most commonly x is called locally compact, if every point x of x has a compact neighbourhood, i. And from my point of view as a settheoretic topologist, the irrationals are far more simpler and more natural than any function space. Furthermore, the intersection of a finite or countably infinite family of locally compact, bcompact subspaces of a given space is bcompact. Compacti cations of symmetric and locally symmetric spaces.
Locally compact hausdorff spaces in this section ucsd math. Let xbe a locally compact space, let kbe a compact set in x, and let dbe an open subset, with k. Show that xis not hausdor, that a is compact, but that ais not closed. Locally compact space an overview sciencedirect topics. Suppose u is open in a locally compact hausdor space x, k.
They have been placed in a separate section because they are not immediately related to the subject matter of the paper. A topological space x is locally compact at point x if there is some. Throughout the chapter, g denotes a locally compact group, i. Every cwcomplex is a compactly generated topological space. The spaces we are concerned with in this presentation are locally compact hausdor spaces. A space is locally compact if it is locally compact at each point. Locally compact space definition of locally compact. This allows one to define integrals of borel measurable functions on g so that. X, there exists an open neighborhood u of x with closure. Chapter 10 compactifications department of mathematics.
A left haar measure on g is a nonzero radon measure on g which is left invariant, this is, xe e for every borel set e. Topology i final exam department of mathematics and. Characterizations of compactness for metric spaces 3 the proof of the main theorem is contained in a sequence of lemmata which we now state. In this book, we give uniform constructions of most known compacti cations of both symmetric and locally symmetric spaces together with some new compacti. Let xbe a locally compact hausdor space, and y a kclosed subset. They are all equivalent if x is a hausdorff space or preregular. A metric space x is sequentially compact if every sequence of points in x has a convergent subsequence converging to a point in x. A path in a topological space x is a continuous map of some closed bounded interval of the real line to the space x. If v is a neighbourhood of xthen so is v\k, and x2y, so v\k\y 6 this shows that x2k\y. The central fact for the theory is the existence of a nonzero leftinvariant radon measure.
If xis locally compact and hausdor, then all compact sets in xare closed and hence if nis a compact neighborhood of xthen ncontains the closure the open intn around x. Every point is contained in a relatively compact open neighborhood. Note that every compact space is locally compact, since the whole space x. In calgebras and their automorphism groups second edition, 2018. Then there exists a compact subspace of q containing a basic neighborhood a,b. X, there exists an open neighborhood u of x with closure u. Proper metrics on locally compact groups, and proper a. A metric space is sequentially compact if and only if every in. If x is locally compact at each of its points, set x is locally compact.
The space xis locally compact if each x2xadmits a compact neighborhood n. The characters form a group gb under pointwise multiplication just as for. X with x 6 y there exist open sets u containing x and v containing y such that u t v 3. Locally compact space that is not topologically complete. The third section contains statements and proofs of four lemmas. A locally compact space is a hausdorff topological space with the property. The integral let t be a locally compact separated space. Y are topological spaces the compactopen topology on x. In mathematics, a locally compact group is a topological group g for which the underlying topology is locally compact and hausdorff. An affirmative answer to this question is apparently cited in the otherwise very useful and complete survey paper ma. Every locally compact space is compactly generated. Clearly, compact spaces and closed subspaces of locally compact.
Locally compact space definition, a topological space in which each point has a neighborhood that is compact. Results about these spaces in general are proved in section 4. We make this into a topological space by using the compactopen topology. Suppose xis a locally compact hausdor space, is a radon measure on x, and is complete. Every point is contained in an open set, that is contained in a closed, compact subset. A metric space is called the space with inner metric, if the distance between any two its points is. Bounded measures in topological spaces by fremlin, garling and haydon proc.
X is a locally compact hausdor space, being the cartesian product of two such. It can be succinctly described as the finest locally convex topology which agrees with that of compact convergence on the unit ball for the supremum norm and the dual is the space of bounded radon measures. Def an inhabited metric space x is locally compact, if each bounded subset can be included in some compact subset of x. To provide an interpretation in terms of irreducible representations of separable, type i groups, a duality theorem and bochner theorem are presented. The concept of a complexification of a locally compact group is defined and its connections with the differential structure developed. Recall separated is equivalent to hausdor a separated space is locally compact if every point has a compact neighborhood. As xis locally compact, xhas a neighbourhood usuch that k uis compact. For every subset e xand every s2s, let e s r be the section of ede ned by e s fx2r. A topological space x is locally compact at point x if there is some compact subspace x of x that contains a neighborhood of x. Thespacez is a zerodimensional locally compact space. Using results announced by stevo todorcevic we establish that if it is consistent that there is a supercompact cardinal then it is consistent that every locally compact perfectly normal space is paracompact. A pseudometric d on a topological space x is proper if its balls are relatively compact.
In the subsequent sections we discuss the proof of the lemmata. Fedorchuk originator, which appeared in encyclopedia of mathematics isbn 1402006098. Proof this followsimmediately from part b of the theorem n corollary 2. Then there is an open set v with compact closure s. In topology and related branches of mathematics, a topological space is called locally compact. Let ct be the set of all continuous, complex valued functions f. Of a locally compact group by kelly mckennon abstract. Big locally compact spaces can be pathological in subtle ways. A of open sets is called an open cover of x if every x. In this section will always be a topological space with topology we are now interested in restrictions on in order to. In fact theyre characterized as the unique topologically complete, nowhere locally compact, separable, zerodimensional, metrizable space. Local compactness is clearly preserved under open continuous maps as open continuous maps preserve both compactness and openness. A summary of the main result of our paper 20 for our current purpose is given by. Ergodic theory is the study of commutative dynamical systems, either in the c.
This article was adapted from an original article by v. Obviously any compact space is paracompact as every open cover admits a finite subcover, let alone a locally finite refinement. Every point is contained in an open set, whose closure is a compact subset. That is, give an example of a topological space xand a subset a. One key feature of locally compact spaces is contained in the following. Every locally compact, second countable g admits a proper a. Let abe a separable calgebra with norm k k a and let bbe a bounded total subset of the selfadjoint part of a. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the haar measure. A topological space is called locally compact if every point has a compact neighbourhood or rather, if one does not at the same time assume that the space is hausdorff topological space, then one needs to require that these compact neighbourhoods exist in a controlled way, e. Note that every compact space is locally compact, since the whole space xsatis es the necessary condition. A topological space is locally compact if every point has an open nbhd with compact closure. A character of a locally compact abelian group g is a continuous group homomorphism from g to s1. Rieszs representation theorem for nonlocally compact spaces. Urysohns lemma suppose x is a locally compact hausdor space, v is open in x, k.773 1198 503 749 124 198 95 563 252 552 628 725 1101 972 538 1275 13 132 356 388 1408 303 1232 1288 46 553 247 156 267 1010 338 1009 493 749 1277 395 734 215 1047 745 322 616 29 1446 801