Complete regularity as a separation axiom

1973 ◽  
Vol 4 (1) ◽  
pp. 39-44 ◽  
Author(s):  
P. Hamburger
Keyword(s):  
Separation Axiom ◽  
Complete Regularity

Complete Regularity as a Separation Axiom

1969 ◽  
Vol 21 ◽  
pp. 96-105 ◽  
Author(s):  
J. de Groot ◽  
J. M. Aarts
Keyword(s):  
Separation Axiom ◽  
Complete Regularity ◽  
Separation Axioms ◽  
The Family

Although the axiom of complete regularity ought to be a separation axiom, in none of its usual forms does it look like an intrinsic separation axiom. Our aim in this paper is to establish such characterizations of complete regularity which naturally fit in between regularity and normality and which already have proved to be fundamental and useful. This can simply be achieved by replacing the family of all open sets (as used in the formulation of the separation axioms) by some suitable (sub)base of open sets. For the sake of simplicity, we assume all our spaces to be T1 and we shall operate with (sub)bases of closed sets (instead of open sets). It is appropriate to define the notion of a screening.Two subsets A and B of a set X are said to be screened by the pair (C, D) if C ∪ D = X, A ∩ D = ∅ and C ∩ B = 0. (Consequently, A ⊂ C and B ⊂ D.)

scholarly journals Pairwise complete regularity as a separation axiom

1990 ◽  
Vol 48 (2) ◽  
pp. 235-245 ◽  
Author(s):  
J. M. Aarts ◽  
M. Mršević
Keyword(s):  
Bitopological Space ◽  
Separation Axiom ◽  
Complete Regularity ◽  
Closed Sets ◽  
Bitopological Spaces ◽  
Natural Way

AbstractFocussing on complete regularity, we discuss the separation properties of bitopological spaces. The unifying concept is that of separation by a pair of bases (B1, B2) for the closed sets of a bitopological space (S, J1, J2). For various separation properties a characterization is presented in terms of separation by a pair of closed bases. This is extended to results concerning pairs of subbases. Here the notion of screening by pairs of subbases plays a central role and the characterization of complete regularity in a natural way fits in between those of regularity and normality. In the key lemma the relation with quasi-proximities is exhibited.

scholarly journals Complete regularity of Ellis semigroups of -actions

2020 ◽  
pp. 1-17
Author(s):  
MARCY BARGE ◽  
JOHANNES KELLENDONK
Keyword(s):  
Complete Regularity ◽  
Completely Regular ◽  
Ellis Semigroup ◽  
Totally Disconnected

Abstract It is shown that the Ellis semigroup of a $\mathbb Z$ -action on a compact totally disconnected space is completely regular if and only if forward proximality coincides with forward asymptoticity and backward proximality coincides with backward asymptoticity. Furthermore, the Ellis semigroup of a $\mathbb Z$ - or $\mathbb R$ -action for which forward proximality and backward proximality are transitive relations is shown to have at most two left minimal ideals. Finally, the notion of near simplicity of the Ellis semigroup is introduced and related to the above.

scholarly journals A new separation axiom in grill topological spaces

2019 ◽  
Vol S (01) ◽  
pp. 706-709
Author(s):  
Maragatha Meenakshi P. ◽  
Chandran S.
Keyword(s):  
Topological Spaces ◽  
Separation Axiom

Complete Regularity

2021 ◽  
pp. 107-136
Author(s):  
Jorge Picado ◽  
Aleš Pultr
Keyword(s):  
Complete Regularity

scholarly journals Topologies on Z n that Are Not Homeomorphic to the n-Dimensional Khalimsky Topological Space

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1072 ◽  
Author(s):  
Sang-Eon Han ◽  
Saeid Jafari ◽  
Jeong Kang
Keyword(s):  
Applied Sciences ◽  
Topological Space ◽  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Discrete Topology ◽  
Separation Axiom ◽  
Different Types

The present paper deals with two types of topologies on the set of integers, Z : a quasi-discrete topology and a topology satisfying the T 1 2 -separation axiom. Furthermore, for each n ∈ N , we develop countably many topologies on Z n which are not homeomorphic to the typical n-dimensional Khalimsky topological space. Based on these different types of new topological structures on Z n , many new mathematical approaches can be done in the fields of pure and applied sciences, such as fixed point theory, rough set theory, and so on.

A general theory of separation, regularity, and connectedness properties

1976 ◽  
Vol 80 (1) ◽  
pp. 81-90 ◽  
Author(s):  
D. M. G. McSherry
Keyword(s):  
General Theory ◽  
Regularity Conditions ◽  
Binary Relations ◽  
Separation Axiom ◽  
Separation Axioms ◽  
The Family

AbstractThe notion of basic topological binary relations is introduced, and corresponding to each such relation we define -regularity, -connectedness, and a separation axiom . The family of separation axioms so obtained is shown to include all the standard axioms T0, T1, …, T5, while familiar properties such as hyperconnectedness and ultraconnectedness are among the class of connectedness conditions. The regularity conditions include R0, R1, and regularity itself.

How to Quantify 'Small-World Networks'?

Fractals ◽  
1998 ◽  
Vol 06 (04) ◽  
pp. 301-303 ◽  
Author(s):  
Hanspeter Herzel
Keyword(s):  
Small World ◽  
Small World Network ◽  
Small Worlds ◽  
Small World Networks ◽  
Complete Regularity ◽  
Local Clustering ◽  
Regular Network ◽  
Analytical Expressions ◽  
Global Connection

Recently Watts and Strogatz emphasized the widespread relevance of 'small worlds' and studied numerically networks between complete regularity and complete randomness. In this letter, I derive simple analytical expressions which can reproduce the empirical observations. It is shown how a few random connections can turn a regular network into a 'small-world network' with a short global connection but persisting local clustering.

scholarly journals A note on the comparison of topologies

2001 ◽  
Vol 26 (4) ◽  
pp. 233-237
Author(s):  
Martin M. Kovár
Keyword(s):  
Separation Axiom ◽  
Covering Properties ◽  
Considerable Problem ◽  
Hausdorff Separation

A considerable problem of some bitopological covering properties is the bitopological unstability with respect to the presence of the pairwise Hausdorff separation axiom. For instance, if the space is RR-pairwise paracompact, its two topologies will collapse and revert to the unitopological case. We introduce a new bitopological separation axiomτS2σwhich is appropriate for the study of the bitopological collapse. We also show that the property that may cause the collapse is much weaker than some modifications of pairwise paracompactness and we generalize several results of T. G. Raghavan and I. L. Reilly (1977) regarding the comparison of topologies.

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