International Journal of Pure and Applied Mathematics
Volume 104 No. 2 2015, 237-247
ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version)
url: http://www.ijpam.eu
doi: http://dx.doi.org/10.12732/ijpam.v104i2.8
AP
ijpam.eu
SOFT TOPOLOGICAL QUESTIONS AND ANSWERS
M. Matejdes
Department of Mathematics and Computer Science
Faculty of Education, Trnava University in Trnava
Priemyselná 4, 918 43 Trnava, SLOVAKIA
Abstract: The paper deals with a few questions concerning a soft topological
space. The main goal is to point out that any soft topological space is homeo-
morphic to a topological space (A × X, τA×X ) where τA×X is a topology on the
product A × X, consequently many soft topological notions and results can be
derived from general topology.
AMS Subject Classification: 54C60, 26A15, 26E25
Key Words: soft set, soft topological space, soft closure, θ-closure, separation
axioms, soft e-continuity, soft e-θ-continuity
1. Introduction
The recent interest (see the references) in the soft topological spaces is growing
and intensive study contributes both to the development of the soft set theory,
but also brings many open problems.
In [5], for a soft topological space (A, X, τ, ), the next propositions were
proved (for the definitions and notations see [5]) and the authors ask if the
converses of Propositions 3.11, 3.16, 3.17, 5.33 below are true as well as they
ask to find a connection between two soft topologies τ and τθ .
c 2015 Academic Publications, Ltd.
Received: July 14, 2015 url: www.acadpubl.eu
, 238 M. Matejdes
(1) Proposition 3.11 If for every x ∈ X, for every a ∈ A and for every a-soft
open neighborhood (F, A) of x there exists an a-soft open neighborhood
(G, A) of x such that x ∈ G(a) ⊂ cl(G, a) ⊂ F (a), then (X, τ, A) is a soft
T3 -space.
(2) Proposition 3.16 Let (F, A) ∈ SS(X, A) and a ∈ A. If there exists a
net S = {xλ , λ ∈ Λ} of X such that xλ ∈a (F, A), for every λ ∈ Λ and x ∈
s-lim(S), then x ∈a cl(F, A).
(3) Proposition 3.17 Let (A, X, τX ) and (B, Y, τY ) be two soft topological
spaces, x ∈ X and e a map of A onto B. If the map f : X → Y is soft e-
continuous at the point x, then for every net S = {xλ , λ ∈ Λ} of X which
soft converges to x in (A, X, τX ) we have that the net {f (xλ ), λ ∈ Λ} of
Y soft converges to f (x) in (B, Y, τY ).
(4) Proposition 5.33 If the map f : X → Y is soft e-θ-continuous, then
Φ−1 −1
f e (G, B) is a soft subset of intθ (Φf e (clθ (G, B)) for every (G, B) ∈ τY .
(5) Under which conditions does the equality τ = τθ holds?
(6) Under what conditions does the sequence τ, τθ , (τθ )θ , ((τθ )θ )θ ,... is even-
tually constant?
(7) Find a soft topological space such that the sequence τ, τθ , (τθ )θ , ((τθ )θ )θ ,...
is strictly decreasing?
2. Topological and soft topological space
Any subset S of the Cartesian product A × X is called a relation from A to
X. By R(A, X), we denote the set of all binary relations from A to X and
S[a] := {x ∈ X : [a, x] ∈ S}. The operations of the sum S ∪ T , ∪t∈T St ,
the intersection S ∩ T , ∩t∈T St , the complement S c and the difference S \ T of
relations are defined obvious way as in the set theory.
By F : A → 2X we denote a set valued mapping from A to the power set
2X of X. A set of all set valued mappings from A to 2X is denoted by F(A, X).
If F, G are two set valued mappings, then F ⊂ G (F = G) means F (a) ⊂ G(a)
(F (a) = G(a)) for any a ∈ A.
A graph of a set valued mapping F is the set Gr(F ) := {[a, x] ∈ A × X :
x ∈ F (a)} and it is a subset of A × X, hence Gr(F ) ∈ R(A, X). So, any set
