Scattered space
Scattered spaces have their origin in Cantors investigations into the uniqueness of trigonometric series. His theorem reads (in modern terminology): If the partial sums of a trigonometric series
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converge to zero except possibly on a set of points of finite scattered height, then all coefficients of the series must be zero.
Scattered spaces and their scattered height are defined as follows. One first defines for any space a transfinite sequence
of subspaces: Let
, for any ordinal
, let
be the derived set of
, and if
is a limit ordinal, put
.
There is a first ordinal for which
. If this derived set
is empty, one calls
a scattered space and the ordinal
its scattered height.
If is compact, then it is readily seen that
must be a successor ordinal, say
. The set
is finite. It is a classical result of S. Mazurkiewicz and J. Sierpiński that a countable compact scattered space
is determined completely by the ordinal
and the number
of points in
:
is homeomorphic to the set of ordinal numbers less than or equal to
.
Compact scattered spaces correspond, via Stone duality, to the so-called superatomic Boolean algebras; these are defined to be those algebras for which every (non-trivial) homomorphic image has an atom. Because of this duality one may say that the structure of compact scattered spaces is understood best.
An important family of scattered spaces (of height ) is constructed as follows: Take an infinite set
and a family
of countably-infinite subsets that is almost disjoint, i.e., if
, then
is finite. The union
is topologized by declaring
to be an open discrete subspace and giving an element
of
basic neighbourhoods of the form
, where
is a finite subset of
. By varying the family
one can obtain various interesting examples of topological spaces, for example, in this way one can make a pseudo-compact space that is not countably compact (cf. Countably-compact space).
References
[a1] | G. Cantor, "Über die Ausdehnung eines Satzes aus der Theorie der trigonometrischen Reihen" Math. Ann. , 5 (1872) pp. 123–132 |
[a2] | S. Mŕowka, "On completely regular spaces" Fund. Math. , 41 (1954) pp. 105–106 |
[a3] | J. Roitman, "Superatomic Boolean algebras" J.D. Monk (ed.) R. Bonnet (ed.) , Handbook of Boolean algebras , 1–3 , North-Holland (1989) pp. Chapt. 19; pp. 719–740 |
Scattered space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Scattered_space&oldid=42510