Page 686 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 686
TOPOLOGY
Chain connected pair of a topological space and its subspace
Zoran Misajleski, misajleski@gf.ukim.edu.mk
Chair of mathematics and informatics, Faculty of Civil Engeneering,
Ss. Cyril and Methodius University in Skopje, North Macedonia
Coauthors: Nikita Shekutkovski, Emin Durmishi, Aneta Velkoska
In this talk we introduce the notion of chain connected set in a topological space given in [1,2,3].
A set C in a topological space X is chain connected in X if for every two elements x and y
and every open covering U of X in X, there exists a chain in U that connects x and y. And
by a chain in U that connects x and y we understand a finite sequence of elements of U such
that x belongs to first element, y to the last, and any two consecutive elements from the chain
have nonempty intersection. Also we introduced the notions of pair of chain separated sets [1]
and weakly chain separated sets [3] in a space. The nonempty sets A and B in a topological
space X are chain separated in X, if there exists an open covering U of X in X such that
for every point a ∈ A and every b ∈ B, there is no chain in U that connects x and y. The
nonempty sets A and B in a topological space X are weakly chain separated in X, if for every
point a ∈ A and every b ∈ B, there exists an open covering U of X in X, such that there is no
chain in U that connects x and y. Clearly, if two sets are chain separated in a topological space,
then they are weakly chain separated in the same space. We give an example of weakly chain
separated sets in a topological space which are not chain separated in the same space. Then
we study the properties of these sets. Moreover we give a criteria for chain connected set in a
topological space by using the notions of chain separatedness and weakly chain separatedness.
A set C is chain connected in a topological space X if and only if it cannot be represented as a
union of two chain (weakly chain) separated sets in X. Then we prove the properties of chain
connected sets in a topological space by using the notions of chain separatedness and weakly
chain separatedness. Furthermore, we give the criteria for two types of topological spaces using
the notion of chain. The topological space is totally separated if any two different singletons are
weakly chain separated in the space, and it is the discrete if they are chain separated. Moreover,
we generalize the notion to a set in a topological space called totally chain separated set. A set
C in a topological space X is totally chain separated in X if every pair of singletons of C are
weakly chain separated in X. At the end we prove the properties of totally chain separated sets
in a topological space. As a consequence, the properties of totally separated spaces are proven
using the notion of chain.
References
[1] Z. Misajleski, N. Shekutkovski, A. Velkoska, Chain Connected Sets In A Topological
Space, Kragujevac Journal of Mathematics, Vol. 43 No. 4, 2019, Pages 575-586;
[2] N. Shekutkovski, Z. Misajleski, E. Durmishi, Chain Connectedness, AIP Conference Pro-
ceedings, Vol. 2183, 030015-1-030015-4, (2019);
[3] N. Shekutkovski, Z. Misajleski, A. Velkoska, E. Durmishi, Weakly Chain Connected Set
In A Topological Space, In a reviewing process.
684
Chain connected pair of a topological space and its subspace
Zoran Misajleski, misajleski@gf.ukim.edu.mk
Chair of mathematics and informatics, Faculty of Civil Engeneering,
Ss. Cyril and Methodius University in Skopje, North Macedonia
Coauthors: Nikita Shekutkovski, Emin Durmishi, Aneta Velkoska
In this talk we introduce the notion of chain connected set in a topological space given in [1,2,3].
A set C in a topological space X is chain connected in X if for every two elements x and y
and every open covering U of X in X, there exists a chain in U that connects x and y. And
by a chain in U that connects x and y we understand a finite sequence of elements of U such
that x belongs to first element, y to the last, and any two consecutive elements from the chain
have nonempty intersection. Also we introduced the notions of pair of chain separated sets [1]
and weakly chain separated sets [3] in a space. The nonempty sets A and B in a topological
space X are chain separated in X, if there exists an open covering U of X in X such that
for every point a ∈ A and every b ∈ B, there is no chain in U that connects x and y. The
nonempty sets A and B in a topological space X are weakly chain separated in X, if for every
point a ∈ A and every b ∈ B, there exists an open covering U of X in X, such that there is no
chain in U that connects x and y. Clearly, if two sets are chain separated in a topological space,
then they are weakly chain separated in the same space. We give an example of weakly chain
separated sets in a topological space which are not chain separated in the same space. Then
we study the properties of these sets. Moreover we give a criteria for chain connected set in a
topological space by using the notions of chain separatedness and weakly chain separatedness.
A set C is chain connected in a topological space X if and only if it cannot be represented as a
union of two chain (weakly chain) separated sets in X. Then we prove the properties of chain
connected sets in a topological space by using the notions of chain separatedness and weakly
chain separatedness. Furthermore, we give the criteria for two types of topological spaces using
the notion of chain. The topological space is totally separated if any two different singletons are
weakly chain separated in the space, and it is the discrete if they are chain separated. Moreover,
we generalize the notion to a set in a topological space called totally chain separated set. A set
C in a topological space X is totally chain separated in X if every pair of singletons of C are
weakly chain separated in X. At the end we prove the properties of totally chain separated sets
in a topological space. As a consequence, the properties of totally separated spaces are proven
using the notion of chain.
References
[1] Z. Misajleski, N. Shekutkovski, A. Velkoska, Chain Connected Sets In A Topological
Space, Kragujevac Journal of Mathematics, Vol. 43 No. 4, 2019, Pages 575-586;
[2] N. Shekutkovski, Z. Misajleski, E. Durmishi, Chain Connectedness, AIP Conference Pro-
ceedings, Vol. 2183, 030015-1-030015-4, (2019);
[3] N. Shekutkovski, Z. Misajleski, A. Velkoska, E. Durmishi, Weakly Chain Connected Set
In A Topological Space, In a reviewing process.
684
