zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
New topologies from old via ideals. (English) Zbl 0723.54005

Let (X,τ) be a topological space, with an ideal of subsets of X. Then β ():={UI: I} is a basis of open sets for a finer topology τ () on X. The authors explore this time-honored method of refining topologies, surveying past results, proving some new results, and improving on old ones. The treatment is tutorial in nature, and includes many examples. As the authors suggest, the paper is suitable for use as a supplement to a general topology course.

The first three sections of the paper deal with the closure and derived set operators for τ (). (Actually this topology is introduced using a closure operator.) In Section 4, O. Njåstad’s notion of “compatibility” is introduced: The topology τ is compatible with the ideal (τ) if A whenever it is the case that for all xA, UA for some neighborhood U of x. The authors show that τ whenver is the ideal of τ-nowhere dense subsets of X, and recast the Banach category theorem (that any union of meager open sets is meager) as the statement that τ whenever is the ideal of 𝒯-meager subsets of X. A nice result is that τ is a hereditarily Lindelöf topology iff τ is compatible with the ideal of countable subsets of X. Also there is Njåstad’s result that β ()=τ() whenever τ·

A highlight of Section 5 is G. Freud’s generalization of the Cantor-Bendixson theorem (that any second countable (even hereditarily Lindelöf) space is the union of a perfect subset and a countable subset), namely: If τ and contains the singleton subsets of X, then every τ ()-closed subset is the union of a τ-perfect set and a set that is in . Also the authors prove that τ and contains all the singletons iff all τ ()-scattered subsets of X are in ·

In Section 6 the authors consider the case when no nonempty τ-open set is in , and show that this condition implies that both τ and τ () have the same semiregularization. The authors also state P. Samuels’ theorem that, under this condition, a function from X to a regular space is continuous with respect to τ iff that function is continuous with respect to τ ()·

Finally, in Section 7 there are some applications. One such is the ease with which “anticompact” spaces (those containing no infinite compact subsets) can be produced. For example, if τ is a Hausdorff topology compatible with , and if contains all the singleton subsets of X, then τ () is anticompact. Other applications involving continuity and θ-continuity are also given.

54A10Several topologies on one set