Abstract. it is shown for a proper closed locally compact

In mathematicsa metric space is a set together with a metric on the set. The metric is a function that defines a concept of distance between any two members of the set, which are usually called points. The metric satisfies a few simple properties. A metric on a space induces topological properties like open and closed setswhich lead to the study of more abstract topological spaces.

The most familiar metric space is 3-dimensional Euclidean space. In fact, a "metric" is the generalization of the Euclidean metric arising from the four long-known properties of the Euclidean distance.

The Euclidean metric defines the distance between two points as the length of the straight line segment connecting them.

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Other metric spaces occur for example in elliptic geometry and hyperbolic geometrywhere distance on a sphere measured by angle is a metric, and the hyperboloid model of hyperbolic geometry is used by special relativity as a metric space of velocities. This is deduced as follows:. Ignoring mathematical details, for any system of roads and terrains the distance between two locations can be defined as the length of the shortest route connecting those locations. To be a metric there shouldn't be any one-way roads.

The triangle inequality expresses the fact that detours aren't shortcuts. If the distance between two points is zero, the two points are indistinguishable from one-another. Many of the examples below can be seen as concrete versions of this general idea. Every metric space is a topological space in a natural manner, and therefore all definitions and theorems about general topological spaces also apply to all metric spaces.

These open balls form the base for a topology on Mmaking it a topological space. The complement of an open set is called closed.

A topological space which can arise in this way from a metric space is called a metrizable space. Equivalently, one can use the general definition of convergence available in all topological spaces.

abstract. it is shown for a proper closed locally compact

Every Euclidean space is complete, as is every closed subset of a complete space. Every metric space has a unique up to isometry completionwhich is a complete space that contains the given space as a dense subset.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

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It only takes a minute to sign up. Thus proved.

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But the problem is it has been given that only continuity will not do the map has to be open also. Thus we get locally compactness. The openness condition is required. Sign up to join this community. The best answers are voted up and rise to the top.

Home Questions Tags Users Unanswered. Asked 5 years, 2 months ago. Active 12 months ago. Viewed 3k times. Please find mistakes in the proof if it exists. Learnmore Learnmore What is the problem? Active Oldest Votes. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Featured on Meta. Feedback post: New moderator reinstatement and appeal process revisions.

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Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I am beginning to learn about compactly generated spaces. Sign up to join this community. The best answers are voted up and rise to the top.

Locally compact space

Home Questions Tags Users Unanswered. Asked 1 month ago. Active 1 month ago. Viewed 35 times.

Metric space

Martin Sleziak Seinkel Seinkel 61 3 3 bronze badges. Active Oldest Votes. Brian M. Scott Brian M. Scott k 43 43 gold badges silver badges bronze badges.

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abstract. it is shown for a proper closed locally compact

Feedback post: New moderator reinstatement and appeal process revisions. The new moderator agreement is now live for moderators to accept across the…. The unofficial elections nomination thread. Hot Network Questions. Question feed.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I know that this proof cannot require much more than a basic topological argument.

abstract. it is shown for a proper closed locally compact

But there's just something that I'm missing. Sign up to join this community. The best answers are voted up and rise to the top.

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Home Questions Tags Users Unanswered. Asked 6 years, 10 months ago. Active 2 years, 11 months ago. Viewed 7k times. Hints or solutions are greatly appreciated. Martin Sleziak Active Oldest Votes. Ayman Hourieh Ayman Hourieh Am I missing something?

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Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.In mathematicsa function between topological spaces is called proper if inverse images of compact subsets are compact. In algebraic geometrythe analogous concept is called a proper morphism. There are several competing descriptions. For instance, a continuous map f is proper if it is closed with compact fibersi.

The two definitions are equivalent if Y is locally compact and Hausdorff. Since the latter is assumed to be compact, it has a finite subcover. Its image is closed in Y, because f is a closed map. Hence the set. If X is Hausdorff and Y is locally compact Hausdorff then proper is equivalent to universally closed. It is possible to generalize the notion of proper maps of topological spaces to locales and topoisee Johnstone From Wikipedia, the free encyclopedia.

This article is about the concept in topology. For the concept in convex analysissee proper convex function. Partial proof of equivalence.

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Proceedings of the American Mathematical Society. Categories : Continuous mappings. Namespaces Article Talk. Views Read Edit View history. Help Community portal Recent changes Upload file. Download as PDF Printable version.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. I seek a proof that is as self contained as possible. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. How to prove that a compact set in a Hausdorff topological space is closed? Ask Question.

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Asked 8 years, 8 months ago. Active 8 years, 8 months ago. Viewed 16k times. Thank you. MathOverview MathOverview Active Oldest Votes. Brian M. Scott Brian M.

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Scott k 43 43 gold badges silver badges bronze badges. Thanks, Brian. Scott Nov 18 '11 at Scott Aug 7 '16 at Mark Mark 4, 26 26 silver badges 29 29 bronze badges. I like it.

abstract. it is shown for a proper closed locally compact

Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name.In topology and related branches of mathematicsa topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.

Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhoodi. There are other common definitions: They are all equivalent if X is a Hausdorff space or preregular. But they are not equivalent in general:.

Condition 1 is probably the most commonly used definition, since it is the least restrictive and the others are equivalent to it when X is Hausdorff. This equivalence is a consequence of the facts that compact subsets of Hausdorff spaces are closed, and closed subsets of compact spaces are compact.

Spaces satisfying 22'2" are usually called strongly locally compact. Condition 4 is used, for example, in Bourbaki. These locally compact Hausdorff LCH spaces are thus the spaces that this article is primarily concerned with.

Every compact Hausdorff space is also locally compact, and many examples of compact spaces may be found in the article compact space. Here we mention only:. As mentioned in the following section, if a Hausdorff space is locally compact, then it is also a Tychonoff space ; there are some examples of Hausdorff spaces that are not Tychonoff spaces in that article. But there are also examples of Tychonoff spaces that fail to be locally compact, such as:. The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section.

The last example contrasts with the Euclidean spaces in the previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it is finite-dimensional in which case it is a Euclidean space. This example also contrasts with the Hilbert cube as an example of a compact space; there is no contradiction because the cube cannot be a neighbourhood of any point in Hilbert space.

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Every locally compact preregular space is, in fact, completely regular. It follows that every locally compact Hausdorff space is a Tychonoff space. Since straight regularity is a more familiar condition than either preregularity which is usually weaker or complete regularity which is usually strongerlocally compact preregular spaces are normally referred to in the mathematical literature as locally compact regular spaces.

Similarly locally compact Tychonoff spaces are usually just referred to as locally compact Hausdorff spaces. Every locally compact Hausdorff space is a Baire space. That is, the conclusion of the Baire category theorem holds: the interior of every union of countably many nowhere dense subsets is empty.

A subspace X of a locally compact Hausdorff space Y is locally compact if and only if X can be written as the set-theoretic difference of two closed subsets of Y. As a corollary, a dense subspace X of a locally compact Hausdorff space Y is locally compact if and only if X is an open subset of Y. Furthermore, if a subspace X of any Hausdorff space Y is locally compact, then X still must be the difference of two closed subsets of Yalthough the converse needn't hold in this case.

Quotient spaces of locally compact Hausdorff spaces are compactly generated. Conversely, every compactly generated Hausdorff space is a quotient of some locally compact Hausdorff space.

For locally compact spaces local uniform convergence is the same as compact convergence. But in fact, there is a simpler method available in the locally compact case; the one-point compactification will embed X in a compact Hausdorff space a X with just one extra point. The one-point compactification can be applied to other spaces, but a X will be Hausdorff if and only if X is locally compact and Hausdorff.

The locally compact Hausdorff spaces can thus be characterised as the open subsets of compact Hausdorff spaces. Intuitively, the extra point in a X can be thought of as a point at infinity.


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