Absolute

Of a regular topological space

The absolute of a regular topological space $ X $ is the space $ aX $ which is mapped perfectly and irreducibly onto $ X $, and is such that any perfect irreducible inverse image of the space $ X $ is homeomorphic to $ aX $. Each regular space $ X $ has a unique absolute. The absolute of a space $ X $ is always extremally disconnected and completely regular, and is perfectly and irreducibly mapped onto $ X $ by means of a transformation $ \pi _ {X} : aX \rightarrow X $. If two spaces $ X $ and $ Y $ are connected by a single-valued or multi-valued perfect irreducible mapping $ f: X \rightarrow Y $, then their absolutes are homeomorphic, and there exists a homeomorphism $ f _ {a} : aX \rightarrow aY $ such that $ f = \pi _ {Y} f _ {a} \pi _ {X} ^ {-1} $.

If a homeomorphism $ f _ {a} : aX \rightarrow aY $ is given, the mapping $ f = \pi _ {Y} f _ {a} \pi _ {X} ^ {-1} $ will be, in the general case, multi-valued, irreducible and perfect. Thus, absolutes and their homeomorphisms "control" the entire class of perfect irreducible mappings of regular spaces. The meaning of this fundamental property is that absolutes of regular topological spaces are projective objects in the category of regular spaces and perfect irreducible mappings. If a regular space $ X $ is compact, respectively, finally compact or complete in the sense of Čech, the respective property is also displayed by the absolute of this space. The absolute of a paracompact space is even strongly paracompact, and is, moreover, perfectly zero-dimensional. However, the absolute of a normal space need not itself be normal. If $ X $ is a completely regular space, then the Stone–Čech compactification of its absolute is the absolute of any compactification of $ X $. Two spaces are called co-absolute if their absolutes are homeomorphic.

Thus, the class of regular spaces is subdivided into disjoint (pairwise not intersecting) classes of co-absolute spaces. A space $ X $ is co-absolute with some metric space if and only if it is a paracompact feathered space containing a dense $ \sigma $-discrete system of open sets. A compact space is co-absolute with some metrizable compactum if and only if it has a countable $ \pi $-weight. If a compact space has a countable $ \pi $-weight and has no isolated points, then and only then it will be co-absolute with the perfect Cantor set. Consequently, all metrizable compacta without isolated points are co-absolute with the perfect Cantor set. The absolute of a countable metrizable compactum is an extension of the Stone–Čech compactification of the natural numbers. The absolute of an extremally disconnected space is homeomorphic to it. Thus, the class of absolutes (whatever this may be) of regular spaces coincides with the class of extremally disconnected spaces. Since a non-discrete extremally disconnected space does not contain any convergent sequence of pairwise distinct points, the absolute of any non-discrete space is non-metrizable (and does not even satisfy the first axiom of countability).

Of the numerous ways in which the absolute $ aX $ of a given (regular) space $ X $ can be constructed, the following method is one of the simplest.

A family $ \xi = \{ A \} $ of non-empty canonical $ \kappa a $-sets, i.e. of canonical closed sets $ A $ of the space $ X $, is called a thread if it is a directed set under inclusion, i.e. if for each two elements $ A, A ^ \prime $ of the family $ \xi $ there exists an element $ A ^ {\prime\prime } $ contained in $ A \cap A ^ \prime $. A thread $ \xi $ is called a maximal or an end thread if it is not a subfamily of any thread different from it. It can be shown that threads exist; moreover, it can be shown that for each non-empty $ \kappa a $-set $ A $ the set $ D _ {A} $ of all threads containing the set $ A $ as an element is non-empty. Each thread is contained in some maximal thread. The intersection of all sets which are elements of a maximal thread $ \xi $ is either empty or contains a single point $ x ( \xi ) $; in the latter case the thread $ \xi $ is said to be convergent (to the point $ x ( \xi ) $). A topology is introduced in the set $ \overline{a} X $ of all ends (i.e. maximal threads), by taking the collection of all sets $ D _ {A} $ as a basis for the closed sets. The resulting topology is Hausdorff and compact. The convergent ends in the compactum $ \overline{a} X $ form an everywhere-dense subspace. The subspace of the space $ \overline{a} X $ consisting of the convergent ends is at the same time the absolute $ aX $ of $ X $; it turns out that $ \overline{a} X $ is identical with the Stone–Čech compactification $ \beta a X $ of $ aX $. If $ X $ is not only regular, but completely regular, the formula of commutativity of the operators $ a $ and $ \beta $ is valid:

$$ \overline{a} X = \beta a X = a \beta X . $$

Comments

The definition of the absolute of a regular topological space $ X $ given above is slightly imprecise. A better (more precise) definition is: The absolute of a regular topological space $ X $ is a pair $ ( a X , \pi _ {X} ) $, where $ \pi _ {X} $ is a perfect irreducible mapping of $ a X $ onto $ X $, such that for every regular topological space $ Y $ and every perfect irreducible mapping $ f $ of $ Y $ onto $ X $ there is a mapping $ g $ of $ a X $ onto $ Y $ such that $ \pi _ {X} = f g $.

In Western literature finally compact spaces are also called Lindelöf spaces and canonical closed set are also called regular closed sets (or sometimes regularly closed sets). The maximal or end threads are commonly called regular closed ultrafilters.

The simple construction of $ \overline{a} X $ given above can be rephrased as follows: The family of regular closed sets forms, in a natural way, a complete Boolean algebra. The space $ \overline{a} X $ is then simply the Stone space of this Boolean algebra (the set of all ultrafilters (cf. Ultrafilter) on it, topologized using the sets $ D _ {A} $ as a base for the closed sets).

References

[a1]	A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian)

Of a $\theta$-proximity space

The $ \theta $-absolute of a $ \theta $-proximity space $ (X, \delta ) $ is the pair $ (X _ \delta , \pi _ {X} ) $ consisting of a proximity space $ X _ \delta $ and a projection $ \pi _ {X} : X _ \delta \rightarrow X $ which is a regular $ \theta $-mapping. Here $ \theta $-mapping is a term denoting any $ \theta $-perfect, irreducible, $ \theta $-proximity-continuous mapping. Any $ \theta $-proximity space has a unique $ \theta $-absolute. Any regular $ \theta $-mapping on a $ \theta $-absolute is a proximity equivalence. The $ \theta $-absolute of a space $ ( X, \delta ) $ is the maximal inverse image of the space $ (X, \delta ) $ under regular $ \theta $-mappings. For each regular $ \theta $-mapping $ f: (X, \delta ) \rightarrow (Y, \delta ^ \prime ) $ there exists a proximity equivalence $ F : X _ \delta \rightarrow Y _ {\delta ^ \prime } $ such that the diagram

$$ \begin{array}{rcl} X _ \delta & \rightarrow ^ { F } &Y _ {\delta ^ \prime } \\ \pi _ {X} \downarrow &{} &\downarrow \pi _ {Y} \\ X & \rightarrow _ { f } & Y \\ \end{array} $$

is commutative.

For maximal $ \theta $-proximities on regular topological spaces the concept of a regular $ \theta $-mapping is identical with that of a perfect irreducible mapping, while the concept of a $ \theta $-absolute is identical with that of the absolute of a regular topological space.

References

[1]	V.I. Ponomarev, "On spaces co-abolute with metric space" Russian Math. Surveys , 21 : 4 (1966) pp. 87–114 Uspekhi Mat. Nauk , 21 : 4 (1966) pp. 101–132
[2]	A.M. Gleason, "Projective topological spaces" Illinois J. Math. , 2 : 4A (1958) pp. 482–489
[3]	V.I. Ponomarev, "Paracompacta, their projection spectra and continuous images" Mat. Sb. , 60 (102) : 1 (1963) pp. 89–119 (In Russian)
[4]	V.V. Fedorchuk, "Perfect irreducible mappings and generalized proximities" Math. USSR-Sb. , 5 : 4 (1968) pp. 498–508 Mat. Sb. , 74 (118) : 4 (1968) pp. 513–536

V.V. Fedorchuk

Of a projective geometry

The absolute in projective geometry is the curve (surface) of the second order constituting the set of infinitely-distant points in the Klein interpretation of a hyperbolic plane (space). The absolute can be used to introduce a metric on a projective plane (space) (cf. Projective determination of a metric). For instance, the projective measure of a segment $ AB $ is defined as a quantity which is proportional to the natural logarithm of the double ratio $ (ABCD) $ of four points, where $ C $ and $ D $ are the points of intersection of the straight line $ AB $ with the absolute.

A.B. Ivanov