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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Cantor spaceMandelbrot space. Then is the set which is open in the Alexandroff extension. But then which is open in the Alexandroff extension.
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. Then c X is open in a compact Hausdorff space so is locally compact Hausdorff, hence its homeomorphic preimage X is also locally compact Hausdorff.
The one-point compactification is usually applied to a non- compact locally compact Hausdorff space. Proof In one direction the statement is that open subspaces of compact Hausdorff spaces are locally compact see there for the proof.
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. Warsaw circleHawaiian earring space. Moreover, if X were compact then c X would be closed in Y and hence not alexandrodf. Retrieved from ” https: Note that a locally compact metric space is not necessarily complete, e.
one-point compactification in nLab
A geometrically appealing example of one-point compactification is given by the inverse stereographic projection.
Checking that this gives us a topology. Real projective space RP n is a compactification of Euclidean space R n. A bit more formally: Email required Address never made public. The product of infinitely many locally compactificaation 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.
The resulting compactification can be thought of as a circle which is compact as a closed and bounded subset of the Euclidean plane. For example, any two different lines in RP 2 intersect in precisely one point, a statement that is not true in R 2.
Topology: One-Point Compactification and Locally Compact Spaces | Mathematics and Such
Under de Morgan duality. Hausdorff spaces are sober.
Then the identity map f: 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. Then each point in X can be identified with an evaluation function on C.
Regarding the first statement: Likewise, can we compactify a general topological space? Thus X can be identified with a subset of compacgification Cthe space of all functions from C to [0,1].
Since the latter is compact by Tychonoff’s theoremthe closure of X as a subset of that space will also be compact.