Alexandroff extension on arvau

Wikipedia / Alexandroff extension

In the mathematical field of topology, the 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. It is named after the Russian mathematician Pavel Alexandroff. More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c:X→X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff space. For such spaces the Alexandroff extension is called the one-point compactification or Alexandroff compactification. The advantages of the Alexandroff compactification lie in its simple, often geometrically meaningful structure and the fact that it is in a precise sense minimal among all compactifications; the disadvantage lies in the fact that it only gives a Hausdorff compactification on the class of locally compact, noncompact Hausdorff spaces, unlike the Stone–Čech compactification which exists for any topological space (but provides an embedding exactly for Tychonoff spaces).

Example: inverse stereographic projection

A geometrically appealing example of one-point compactification is given by the inverse stereographic projection. Recall that the stereographic projection S as K. This example already contains the key concepts of the general case.

Motivation

be an embedding from a topological space X to a compact Hausdorff topological space Y. 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. Moreover, if X were compact then c(X) would be closed in Y and hence not dense. Thus a space can only admit a Hausdorff one-point compactification if it is locally compact, noncompact and Hausdorff. Moreover, in such a one-point compactification the image of a neighborhood basis for x in X gives a neighborhood basis for c(x) in c(X to the image under c of a subset of X with compact complement.

The Alexandroff extension

by taking as open sets all the open sets in X where C is closed and compact in X is compact .

is called the Alexandroff extension of X

The properties below follow from the above discussion:

The map c is continuous and open: it embeds X.
is compact.
The image c(X, if X is noncompact.
is Hausdorff if and only if X is Hausdorff and locally compact.
is T1 if and only if X is T1.

The one-point compactification

is a Hausdorff compactification of X if and only if X is Hausdorff, noncompact and locally compact. In this case it is called the one-point compactification or Alexandroff compactification of X.

is a limit point (i.e. not an isolated point.

Let X be any noncompact Tychonoff space of equivalence classes of compactifications, any minimal element is equivalent to the Alexandroff extension (Engelking, Theorem 3.5.12). It follows that a noncompact Tychonoff space admits a minimal compactification if and only if it is locally compact.

Non-Hausdorff one-point compactifications

obtained by adding a single point, which could also be called one-point compactifications is necessarily an open., as previously discussed.

are as follows:

.
The open extension topology.
; for example, the complements of all finite closed compact subsets, or the complements of all countable closed compact subsets.

Further examples

Compactifications of discrete spaces

The one-point compactification of the set of positive integers is homeomorphic to the space consisting of K = {0} U {1/n | n is a positive integer} with the order topology.
has the discrete topology.
Polyadic spaces are defined as topological spaces that are the continuous image of the power of a one-point compactification of a discrete, locally compact Hausdorff space.

Compactifications of continuous spaces

The one-point compactification of n-dimensional Euclidean space Rn is homeomorphic to the n-sphere Sn. As above, the map can be given explicitly as an n-dimensional inverse stereographic projection.
.
of copies of the interval (0,1) is a n circles.
The one-point compactification of the disjoint union of a countable number of copies of the interval (0,1) is the Hawaiian earring. This is different from the wedge of countably many circles, which is not compact.
, where the forward slash denotes the quotient space.
is the smash product is the wedge sum, and again, / denotes the quotient space.

As a functor

The Alexandroff extension can be viewed as a functor from the category of topological spaces, where interval category is the category with 2 objects connected by single arrow.