This one-point compactification is also known as the Alexandroff compactification after a paper by Павел Сергеевич Александров (then. The one point compactification. Definition A compactification of a topological space X is a compact topological space Y containing X as a subspace. of topological spaces and the Alexandroff one point compactification. Some prop- erties of the locally compact spaces and one point compactification are proved.

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Recall from the above discussion that any compactification with one point remainder is necessarily isomorphic to the Alexandroff compactification.

Home About This Blog Contents. By stereographic projection we have a homeomorphism. The product of infinitely many locally compact spaces is not locally compact in general to get a locally compact space, one uses the restricted product which leads to adeles and ideles in algebraic number theory.

Fill in your details below or click an icon to log in: For example, any two different lines in RP 2 intersect in precisely one point, a statement that is not true in R 2. This entry was posted in Notes and tagged advancedAlexandroff extensionscompact spacescompactificationsHausdorfflocally compact spacesone-point compactificationtopology. If X X is Alexandrofcthen it is sufficient to speak of compact subsets in def. We poijt to show that i: Remark If X X is Hausdorffthen it is sufficient to speak of co,pactification subsets in def.

The inclusion map c: The easiest way is to add just one point. Then each point in X can be identified with an evaluation function on C.


Topology: One-Point Compactification and Locally Compact Spaces | Mathematics and Such

For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. Views Read Edit View history. In particular, the Alexandroff extension c: But then which is open in the Alexandroff extension. The Alexandroff extension can be viewed as a functor from the category of topological spaces with proper continuous maps as morphisms to the category whose coompactification are continuous maps c: Retrieved from ” https: Compactificatio construction presents the J-homomorphism in stable homotopy theory and is encoded for instance in the definition of orthogonal spectra.

Alexandroff extension – Wikipedia

Regarding the first point: Hausdorff spaces are sober. The methods of compactification are various, but each is a way of controlling points from “going off to infinity” by in some way adding “points at infinity” or preventing such an “escape”. Checking that this gives us a topology. For example, modular curves compactificaion compactified by the ponit of single points for each cuspmaking them Riemann surfaces and so, since they are compact, algebraic curves.

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Embeddings into compact Hausdorff spaces may be of particular interest. Of particular interest are Hausdorff compactifications, i. The fact that large and interesting classes of non-compact spaces do in fact have compactifications of particular sorts makes compactification a common technique in topology.

Proposition one-point extension of locally compact space is Hausdorff precisely if original space is Let X X be a locally compact topological space.


We need to show that this has a finite subcover. In particular, homeomorphic compacification have isomorphic Alexandroff extensions. You are commenting using your Twitter account.

The topology on the one-point extension in def. Let be an open cover. Definition one-point extension Let X X be any topological space. Regarding the third point: In the more general situation, it may not really be a compactification and hence is called the one-point extension or Alexandroff extension.

Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems.

Alexandroff extension

Let X X be any topological space. In this context and in view of the previous case, one usually writes. Kolmogorov spaceHausdorff spaceregular spacenormal space.

The one-point compactification of X is Hausdorff if and only if X is Hausdorff and locally compact. Here the alexandrodf are there for a good reason: Let X be any noncompact Tychonoff space.

For each possible “direction” in which points in Compactifocation n can “escape”, one new point at infinity is added but each direction is identified with its opposite.

It is named for the Russian mathematician Pavel Alexandrov.