Alexander's theorem for real-compactness

1968 ◽  
Vol 64 (1) ◽  
pp. 41-43 ◽  
Author(s):  
Allan Hayes
Keyword(s):  
Topological Space ◽  
Intersection Property ◽  
Finite Intersection ◽  
Closed Sets ◽  
Empty Intersection ◽  
Finite Intersection Property

Alexander's theorem (5) states that a topological space is compact if there is a sub-base, , for its closed sets such that every subclass of with the finite intersection property has a non-empty intersection. An analysis and extension of this is given here which has applications, inter alia, to problems concerning real-compactness (2).

scholarly journals Intrinsic characterizations of C-realcompact spaces

2021 ◽  
Vol 22 (2) ◽  
pp. 295
Author(s):  
Sudip Kumar Acharyya ◽  
Rakesh Bharati ◽  
Atasi Deb Ray
Keyword(s):  
Topological Space ◽  
Dense Subspace ◽  
Finite Intersection ◽  
Compact Spaces ◽  
Closed Sets ◽  
Finite Intersection Property ◽  
Arbitrary Topological Space ◽  
Stable Family ◽  
Definition Of

<pre>c-realcompact spaces are introduced by Karamzadeh and Keshtkar in Quaest. Math. 41, no. 8 (2018), 1135-1167. We offer a characterization of these spaces X via c-stable family of closed sets in X by showing that  X is c-realcompact if and only if each c-stable family of closed sets in X with finite intersection property has nonempty intersection. This last condition which makes sense for an arbitrary topological space can be taken as an alternative definition of a c-realcompact space. We show that each topological space can be extended as a dense subspace to a c-realcompact space with some desired extension properties. An allied class of spaces viz CP-compact spaces akin to that of c-realcompact spaces are introduced. The paper ends after examining how far a known class of c-realcompact spaces could be realized as CP-compact for appropriately chosen ideal P of closed sets in X.</pre>

scholarly journals The Hausdorff dimension and exact Hausdorff measure of random recursive sets with overlapping

2002 ◽  
Vol 31 (1) ◽  
pp. 11-21 ◽  
Author(s):  
Hongwen Guo ◽  
Dihe Hu
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Intersection Property ◽  
Random Sets ◽  
Hausdorff Measures ◽  
Open Set Condition ◽  
Finite Intersection ◽  
Open Set ◽  
Finite Intersection Property

We weaken the open set condition and define a finite intersection property in the construction of the random recursive sets. We prove that this larger class of random sets are fractals in the sense of Taylor, and give conditions when these sets have positive and finite Hausdorff measures, which in certain extent generalize some of the known results, about random recursive fractals.

scholarly journals Finite Intersection Property and Dynamical Compactness

2017 ◽  
Vol 30 (3) ◽  
pp. 1221-1245
Author(s):  
Wen Huang ◽  
Danylo Khilko ◽  
Sergiĭ Kolyada ◽  
Alfred Peris ◽  
Guohua Zhang
Keyword(s):  
Intersection Property ◽  
Finite Intersection ◽  
Finite Intersection Property

On a Generalized Finite Intersection Property

2012 ◽  
Vol 40 (6) ◽  
pp. 2151-2160 ◽  
Author(s):  
John Clark ◽  
Yasuyuki Hirano ◽  
Hong Kee Kim ◽  
Yang Lee
Keyword(s):  
Intersection Property ◽  
Finite Intersection ◽  
Finite Intersection Property

scholarly journals Shells, sequences and intersections

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4851-4856
Author(s):  
Luigi Papini
Keyword(s):  
Intersection Property ◽  
Finite Intersection ◽  
Half Century ◽  
Finite Intersection Property ◽  
Increasing Sequences

In this paper we consider two facts concerning shells. First, we deal with ?nested? (decreasing or increasing) sequences of shells. We prove that the intersection, as well as the closure of the union of these sequences, is a shell. Secondly, we consider some questions raised in a paper by Stiles on shells, published half century ago. He left open some questions, also connected with ?spheres? (boundaries of balls), and with a finite intersection property. Here we give a new result on these problems.

Compactification of groups and rings and nonstandard analysis

1970 ◽  
Vol 34 (4) ◽  
pp. 576-588 ◽  
Author(s):  
Abraham Robinson
Keyword(s):  
Topological Group ◽  
Finite Subset ◽  
Nonstandard Analysis ◽  
Nonempty Intersection ◽  
Intersection Property ◽  
Finite Intersection ◽  
Finite Intersection Property ◽  
Formal Properties

Let G be a separated (Hausdorff) topological group and let *G be an enlargement of G (see [8]). Thus, *G (i) possesses the same formal properties as G in the sense explained in [8], and (ii) every set of subsets {Aν} of G with the finite intersection property—i.e. such that every nonempty finite subset of {Aν} has a nonempty intersection—satisfies ∩*Aν ≠ ø, where the *Aν are the extensions of the Aν in *G, respectively.

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