Alexandroff extension explained
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 gives an explicit homeomorphism from the unit sphere minus the north pole (0,0,1) to the Euclidean plane. The inverse stereographic projection
is an open, dense embedding into a compact Hausdorff space obtained by adjoining the additional point
. Under the stereographic projection latitudinal circles
get mapped to planar circles
. It follows that the deleted neighborhood basis of
given by the punctured spherical caps
corresponds to the complements of closed planar disks
. More qualitatively, a neighborhood basis at
is furnished by the sets
S-1(R2\setminusK)\cup\{infty\}
as
K ranges through the compact subsets of
. This example already contains the key concepts of the general case.
Motivation
Let
be an embedding from a topological space
X to a compact Hausdorff topological space
Y, with dense image and one-point remainder
. 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), and—because a subset of a compact Hausdorff space is compact if and only if it is closed—the open neighborhoods of
must be all sets obtained by adjoining
to the image under
c of a subset of
X with compact complement.
The Alexandroff extension
Let
be a topological space. Put
and topologize
by taking as open sets all the open sets in
X together with all sets of the form
V=(X\setminusC)\cup\{infty\}
where
C is closed and compact in
X. Here,
denotes the complement of
in
Note that
is an open neighborhood of
and thus any open cover of
will contain all except a compact subset
of
implying that
is compact .
The space
is called the
Alexandroff extension of
X (Willard, 19A). Sometimes the same name is used for the inclusion map
The properties below follow from the above discussion:
- The map c is continuous and open: it embeds X as an open subset of
.
is compact.
- The image c(X) is dense in
, if
X is noncompact.
is
Hausdorff if and only if
X is Hausdorff and
locally compact.
is
T1 if and only if
X is T
1.
The one-point compactification
In particular, the Alexandroff extension
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.
Recall from the above discussion that any Hausdorff compactification with one point remainder is necessarily (isomorphic to) the Alexandroff compactification. In particular, if
is a compact Hausdorff space and
is a
limit point of
(i.e. not an
isolated point of
),
is the Alexandroff compactification of
.
Let X be any noncompact Tychonoff space. Under the natural partial ordering on the set
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
Let
be an arbitrary noncompact topological space. One may want to determine all the compactifications (not necessarily Hausdorff) of
obtained by adding a single point, which could also be called
one-point compactifications in this context. So one wants to determine all possible ways to give
a compact topology such that
is dense in it and the subspace topology on
induced from
is the same as the original topology. The last compatibility condition on the topology automatically implies that
is dense in
, because
is not compact, so it cannot be closed in a compact space.Also, it is a fact that the inclusion map
is necessarily an
open embedding, that is,
must be open in
and the topology on
must contain every memberof
.
[1] So the topology on
is determined by the neighbourhoods of
. Any neighborhood of
is necessarily the complement in
of a closed compact subset of
, as previously discussed.
The topologies on
that make it a compactification of
are as follows:
- The Alexandroff extension of
defined above. Here we take the complements of all closed compact subsets of
as neighborhoods of
. This is the largest topology that makes
a one-point compactification of
.
, namely the whole space
. This is the smallest topology that makes
a one-point compactification of
.
- Any topology intermediate between the two topologies above. For neighborhoods of
one has to pick a suitable subfamily of the complements of all closed compact subsets of
; 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 = U with the order topology.
- A sequence
in a topological space
converges to a point
in
, if and only if the map
given by
for
in
and
is continuous. Here
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.
- The one-point compactification of the product of
copies of the half-closed interval [0,1), that is, of <math>[0,1)^\kappa</math>, is (homeomorphic to) <math>[0,1]^\kappa.
- Since the closure of a connected subset is connected, the Alexandroff extension of a noncompact connected space is connected. However a one-point compactification may "connect" a disconnected space: for instance the one-point compactification of the disjoint union of a finite number
of copies of the interval (0,1) is a
wedge of
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.
- Given
compact Hausdorff and
any closed subset of
, the one-point compactification of
is
, where the forward slash denotes the
quotient space.
[2]
and
are locally compact Hausdorff, then
where
is the
smash product. Recall that the definition of the smash product:
A\wedgeB=(A x B)/(A\veeB)
where
is the
wedge sum, and again, / denotes the quotient space.
[2] As a functor
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 objects are continuous maps
and for which the morphisms from
to
are pairs of continuous maps
fX\colonX1 → X2, fY\colonY1 → Y2
such that
. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.
Notes and References
- Web site: General topology – Non-Hausdorff one-point compactifications.
- [Joseph J. Rotman]