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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In the mathematical field of topologythe Alexandroff extension is a way to extend a noncompact topological space by adjoining a single point in such a way that the resulting space is compact. Note that a locally compact metric space is not necessarily complete, e. For example, compact spaces are clearly locally compact. The easiest way is to add just one point.
Regarding the second point: The inclusion map c: Of particular interest are Hausdorff compactifications, i. The cusps stand in for those different ‘directions to infinity’.
Views Read Edit View history. The topology on the one-point extension in def.
Alsxandroff If X X is Hausdorffthen it is sufficient to speak of compact subsets in def. Since every compact Hausdorff space is a Tychonoff spaceand every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space. Proof The unions and finite intersections of the open subsets inherited from X X are closed among themselves by the assumption that X X is a topological space. A one-point compactification of compsctification given by the union of two circles which are tangent to each other.
To see why, replacing an open subset by a smaller one, we may take containing 0. Proof In one direction the statement is that open subspaces of compact Hausdorff spaces are locally compact see there for the proof. Passing to projective space is a common tool in algebraic geometry because the added points at infinity lead to simpler formulations of many theorems. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.
Then each point in X can be identified with an evaluation function on C. This follows because subsets are closed in a closed subspace precisely if they are closed in the ambient space and because closed subsets of compact spaces are compact. Every locally compact Hausdorff space is homemorphic to a open topological subspace of a compact topological space.
Alexandroff extension – Wikipedia
Embeddings into compact Hausdorff spaces may be of particular interest. Under de Morgan duality. If X is locally compact, then so is any open subset U. This page was last edited on 23 Octoberat Complex projective space CP n is also a compactification of C n ; the Alexandroff one-point compactification of the plane C is homeomorphic to the complex projective line CP 1which in turn can be identified with a sphere, the Riemann sphere. August 12, at 2: A topological space has a Hausdorff compactification if and only if it is Tychonoff.
This is harder to compactify. Note compactificatiln that the projective plane RP 2 is not the one-point compactification of the plane R 2 since more than one point is added.
It is often useful to embed topological spaces in compact spacesbecause of the special properties compact spaces have. Let X be any noncompact Tychonoff space. Checking that this gives us a topology.
In one direction the statement is obe open subspaces of compact Hausdorff spaces are locally compact see there for the proof. Indeed, a union of sets of the first type is of the first type.
This is notably used in the Deligne—Mumford compactification of the moduli space of algebraic curves.
In particular, a disjoint union of locally compact spaces is alexahdroff compact. It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact. We need to show that i: You are commenting using your Twitter account. Intuitively, the process can be pictured as follows: 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.
Recall from the above discussion that any compactification with one point dompactification is necessarily isomorphic to the Alexandroff compactification.
You are commenting using your Facebook account. Cantor spaceMandelbrot space. The operation of one-point compactification is not a functor on the whole category of topological spaces. Here the cusps are there for a good aexandroff Let X be a Hausdorff space.