Ratio Mathematica Volume 44, 2022
On Intuitionistic Semi * Continuous Functions
G. Esther Rathinakani*
M. Navaneethakrishnan†
Abstract
In this paper we introduce intuitionistic semi * continuous and intuitionistic contra -
semi * continuous functions via the concept of intuitionistic semi * open and
intuitionistic semi * closed set respectively. Also, we investigate their properties and
characterization.
Keywords: intuitionistic semi * open, intuitionistic semi * closed, intuitionistic semi *
continuous, intuitionistic contra-semi * closed.
2010 AMS subject classification: 54C05‡
*Research Scholar, Reg. No. 19222102092011, PG and Research Department of Mathematics, Kamaraj
College, Thoothukudi, Tamilnadu, India. Affiliated to Manonmaniam Sundaranar University,
Abishekapatti, Tirunelveli, Tamilnadu, India; e-mail estherrathinakani@gmail.com.
†Associate Professor, PG and Research Department of Mathematics, Kamaraj College, Thoothukudi,
Tamilnadu, India. Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli,
Tamilnadu, India; e-mail navaneethan65@yahoo.co.in.
‡ Received on June 8th, 2022. Accepted on Aug 10th, 2022. Published on Nov 30th, 2022. doi:
10.23755/rm.v44i0.891. ISSN: 1592-7415. eISSN: 2282-8214. ©The Authors. This paper is published
under the CC-BY licence agreement
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G. Esther Rathinakani and M. Navaneethakrishnan
1. Introduction
The concept of intuitionistic set introduce by D. Coker [2] in 1996 and also he [1]
has introduced the concept of intuitionistic topological space. In [5], we introduced the
concept of intuitionistic generalized closure operator and defined a new intuitionistic
topology τ* and studied their properties. Also, in [6] we introduced a new open set
namely intuitionistic semi * open and studied their properties. In this paper we define
intuitionistic semi * continuous and intuitionistic contra - semi * continuous functions
via the concept of intuitionistic semi * open and intuitionistic semi * closed set
respectively. Also, we investigate their properties and characterization.
2.Preliminaries
Definition 2.1 [1] Let X be a nonempty fixed set. An intuitionistic set (IS in short) Ã is
an object having the form à =< 𝑋, 𝐴1, 𝐴2 > where 𝐴1 and 𝐴2 are subsets of 𝑋 such that
𝐴1 ∩ 𝐴2 = ∅. The set 𝐴1 is called the set of member of à , while 𝐴2 is called the set of
non member of à .
Definition 2.2 [1] An intuitionistic topology (IT in short) by subsets of a nonempty set
X is a family 𝜏 of IS’s satisfying the following axioms.
(a) ∅̃I , XĨ ∈ 𝜏,
(b) G̃1 ∩ G̃2 ∈ 𝜏 for every G̃1 , G̃2 ∈ 𝜏, and
(c) ∪ G̃ 𝑖 ∈ 𝜏 for any arbitrary family {G̃ : 𝑖 ∈ 𝐽} ⊆ 𝜏.
The pair (𝑋, 𝜏) is called an intuitionistic topological space (ITS in short) and any
IS à in 𝜏 is called an intuitionistic open set (IOS). The complement of an IO set à in is
called an intuitionistic closed set (ICS).
Definition 2.3 [1] Let (𝑋, 𝜏) be an ITS and à =< 𝑋, 𝐴1, 𝐴2 > be an IS in X. Then the
interior and the closure of 𝐴 are denoted by Iint(Ã ) and I𝑐𝑙(Ã ), and are defined as
follows.
Iint(Ã ) = ∪ {G̃ | G̃ is an IOS and G̃ ⊆ Ã } and
Icl(à ) = ∩ {K̃ | K̃ is an ICS and à ⊆ K̃}.
Definition 2.4 [2] Let X be a nonempty set and p ∈ X be a fixed element. Then the IS p̃
defined by p̃ =< X, {p}, {p}
c > is called an intuitionistic point (in short, IP).
Definition 2.5 [10] Let (𝑋, 𝜏) be an ITS and à =< 𝑋, 𝐴1, 𝐴2 > be an IS in X, à is said to
be intuitionistic generalized closed set (briefly Ig – closed set ) Icl(Ã ) ⊆ Ũ whenever Ã
⊆ Ũ and Ũ is IOS in X.
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On Intuitionistic Semi * Continuous Functions
Definition 2.6 [5] If �̃� is an IS of an ITS (X, τ), then the intuitionistic generalized
closure of �̃� is defined as the intersection of all Ig – closed sets in X containing �̃� and is
denoted by Icl*(�̃�).
Definition 2.7 [6] The IS �̃� of an ITS (X, τ) is called intuitionistic semi * open sets if
there is an intuitionstic open set �̃� in X such that �̃� ⊆ �̃� ⊆ Icl*(�̃�).
Definition 2.8 [6] The intuitionistic semi * interior of �̃� is defined as the union of all
intuitionistic semi * open sets of X contained in �̃�. It is denoted by IS*int(�̃�).
Definition 2.9 An intuitionistic set �̃� of a ITS (X, τ) is called an intuitionistic semi *
closed set if X - �̃� is intuitionistic semi * open.
Definition 2.10 The semi * closure of an IS �̃� is defined as the intersection of all
intuitionistic semi * closed sets in X that containing �̃�. It is denoted by IS*cl(�̃�).
Theorem 2.11 Let (X, τ) be an ITS and �̃� be an IS of X. Then
(i) IS*cl(X - �̃�) = X – IS*int(�̃�)
(ii) I S*int(X - �̃�) = X – IS*cl(�̃�)
Definition 2.12[9] A function f : X → Y is said to be intuitionistic semi continuous if
𝑓 −1(�̃�) is ISO in X for every IOS �̃� in Y.
Theorem 2.13[6] Let (X, τ) be an ITS and �̃� be an IS of X. Then
(i) Every IOS is IS*O.
(ii) Every IS*O is ISO.
(iii) Every ICS is IS*C.
(iv) Every IS*C is ISC.
Theorem 2.14[6] Let (X, τ) be an ITS. Then
(i) If {�̃�𝛼} is a collection of IS*O in X then ⋃ �̃�𝛼 is IS*O.
(ii) If �̃� is IS*O in X and �̃� is an IOS in X, then �̃� ∩ �̃� is IS*O in X.
Theorem 2.15[6] Let (X, τ) be an ITS and �̃� be an IS of X. Then
(i) �̃� is IS*O if and only if IS*int(�̃�) = �̃�.
(ii) �̃� is IS*C if and only if IS*cl(�̃�) = �̃�.
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G. Esther Rathinakani and M. Navaneethakrishnan
Theorem 2.16[6] Let (X, τ) be an ITS, �̃� be an IS of X and 𝑝 ∈ X. Then 𝑝 ∈ IS*cl(�̃�) if
and only if every IS*O in X containing 𝑝 intersects �̃�.
Definition 2.17[7] Let (X, τ) be an ITS and �̃� be an IS of X. Then the intuitionistic semi
* frontier of �̃� (denoted by IS*Fr (�̃�)) is defined by IS*Fr (�̃�) = IS*cl(�̃�) - IS*int (�̃�).
Theorem 2.18[7] Let (X, τ) be an ITS and �̃� be an IS of X. Then
IS*Fr (�̃�) = IS*cl(�̃�) ⋂ IS*cl (�̃�).
3. Intuitionistic Semi * Continuous Functions
Definition 3.1 A function f : X →Y is said to be intuitionistic semi * continuous at 𝑝 ∈
X if for each intuitionistic open set �̃� of Y containing f(𝑝), there is an intuitionistic semi
open set �̃� in X such that 𝑝 ∈ �̃� and f(�̃�) ⊆ �̃�.
Definition 3.2 A function f : X → Y is said to be intuitionistic semi * continuous if
𝑓 −1(�̃�) is IS*O in X for every IOS �̃� in Y.
Theorem 3.3 Every intuitionistic continuous function is intuitionistic semi *
continuous.
Proof: Let f : X → Y be intuitionistic continuous and �̃� be IO in Y. Then 𝑓 −1(�̃�) is IO
in X. Therefore by theorem 2.13(i), 𝑓 −1(�̃�) is IS*O in X. Hence f is intuitionistic semi *
continuous function.
Remark 3.4 The converse of the above theorem need not be true as seen from the
succeeding example
Example 3.5 Let X = {i, j, k} = Y and τ1= {𝑋�̃�, ∅̃𝐼 , < X, {i},{j, k} >, < X, {j}, {i, k} >, <
X, {i, j}, {k}>}, τ2= {𝑋�̃�, ∅̃𝐼 , < X, {j},{i, k} >, < X, {i}, {j} >, < X, {i, j}, ∅>}. Let f: (X,
τ1) → (Y, τ2) be defined by f(i) = j, f(j) = i, f(k) = k. Then f is intuitionistic semi *
continuous. Let �̃� = < X, {i, j}, ∅>. Then 𝑓 −1(�̃�) = < X, {k, i}, ∅> is not IOS in τ1.
Therefore f is not an intuitionistic continuous.
Corollary 3.6 Every constant function is intuitionistic semi * continuous function.
Proof: We know that every constant function is intuitionistic continuous function.
Therefore by theorem 3.3 every constant function is intuitionistic semi * continuous
function.
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On Intuitionistic Semi * Continuous Functions
Theorem 3.7 Let ẞ be the intuitionistic basis of the intuitionistic topological space Y.
Then the function f: X →Y is intuitionistic semi * continuous if and only if inverse
image of every basic IOS in Y under the function f is IS*O in X.
Proof: Let f: X →Y be intuitionistic semi * continuous. Then the inverse image of every
IOS in Y is IS*O in X. In particular, the inverse image of every basic IOS in Y is IS*O
in X. Coversely, assume that �̃� be an IOS in Y. Then �̃� = ⋃ �̃�α where �̃�α∈ ẞ. Now
𝑓 −1(�̃�) = 𝑓 −1(⋃ �̃�α) = ⋃ 𝑓 −1(�̃�α). Therefore by hypothesis, 𝑓 −1 (�̃�α) is IS*O for each
α. Then by theorem 2.14(i), 𝑓 −1 (�̃�) is IS*O. Hence the function f is intuitionistic semi
* continuous.
Theorem 3.8 Every intuitionistic semi * continuous function is intuitionistic semi
continuous.
Proof: Let f: X →Y be intuitionistic semi * continuous function and �̃� be an IOS in Y.
Then 𝑓 −1(�̃�) is IS*O in X. Therefore by theorem 2.13(ii), 𝑓 −1(�̃�) is ISO in X. Hence f
is intuitionistic semi continuous.
Remark 3.9 The following example shows that the converse of the above theorem need
not be true.
Example 3.10 Let X = {i, j, k} = Y and τ1= {𝑋�̃�, ∅̃𝐼 , < X, {j},{i, k} >, < X, {i}, {j} >, <
X, {i, j}, ∅>}, τ2= {𝑋�̃�, ∅̃𝐼 , < X, {i},{j} >, < X, {i, j}, ∅>}. Let f: (X, τ1) → (Y, τ2) be
defined by f(i) = j, f(j) = i, f(k) = k. Then f is intuitionistic semi continuous. Let �̃� = < X,
{j}, {i}>. Then 𝑓 −1(�̃�) = < X, {j}, {i}> is not an IS*O in τ1. Therefore f is not an
intuitionistic semi * continuous.
Lemma 3.11 Let (X. τ) be an ITS and �̃� be an IS of X. Then
(i) �̃� is IS*O in X if and only if Icl*(Iint(�̃�)) = Icl*(�̃�).
(ii) �̃� is IS*C in X if and only if Iint*(Icl(�̃�)) = Iint*(�̃�).
Proof: (i) Let �̃� be an IS*O. Then by definition of IS*O we have �̃� ⊆ Icl*(Iint(�̃�)).
Hence Icl*(�̃�) ⊆ Icl*(Iint(�̃�)). Also we have Iint(�̃�) ⊆ �̃� , Icl*(Iint(�̃�)) ⊆ Icl*(�̃�). Thus
Icl*(Iint(�̃�)) = Icl*(�̃�). On the other hand, let Icl*(Iint(�̃�)) = Icl*(�̃�). Then by definition
of IS*O, �̃� is IS*O.
(ii) �̃� is IS*C if and only if X - �̃� is IS*O. Then by (i) �̃� is IS*C if and only if
Icl*(Iint(X - �̃�)) = Icl*(X - �̃�). Hence �̃� is IS*C if and only if Iint*(Icl(�̃�)) = Iint*(�̃�).
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G. Esther Rathinakani and M. Navaneethakrishnan
Theorem 3.12 Let f: X → Y be a function. Then the following are equivalent.
(i) f is intuitionistic semi * continuous.
(ii) f is intuitionistic semi * continuous at each IP of X.
(iii) 𝑓 −1(�̃�) is IS*C in X for every ICS �̃� in Y.
(iv) f(IS*cl(�̃�)) ⊆ Icl(f(�̃�)) for every IS �̃� of X.
(v) IS*cl(𝑓 −1 (�̃�)) ⊆ 𝑓 −1(Icl(�̃�)) for every IS �̃� of Y.
(vi) Iint*(Icl(𝑓 −1 (�̃�))) = Iint*(𝑓 −1 (�̃�)) for every ICS �̃� in Y.
(vii) Icl*( Iint(𝑓 −1 (�̃�))) = Icl*(𝑓 −1 (�̃�)) for every IOS �̃� in Y.
(viii) 𝑓 −1 (Iint(�̃�) ⊆ IS*int(𝑓 −1 (�̃�)) for every IS �̃� in Y.
Proof: (i) ⟹ (ii). Let f: X → Y be an intuitionistic semi * continuous. Let 𝑝 ∈ X and �̃�
be an IOS in Y containing f(�̃�). Then 𝑝 ∈ 𝑓 −1(�̃�). Since f is intuitionistic semi *
continuous, �̃� = 𝑓 −1(�̃�) is an IS*O in X containing 𝑝 such that f(�̃�) ⊆ �̃�. Hence f is
intuitionistic semi * continuous at each IP of X.
(ii)⟹ (iii). Let �̃� be an ICS in Y. Then �̃� = Y - �̃� is an IOS in Y. Let 𝑝 ∈ 𝑓 −1(�̃�). Then
f(𝑝) ∈ �̃�. By hypothesis, there is a IS*O set �̃�𝑝 in X containing 𝑝 such that f(𝑝) ∈ f(�̃�𝑝)
⊆ �̃�. Therefore �̃�𝑝 ⊆ 𝑓
−1(�̃�). Hence 𝑓 −1(�̃�) = ∪ {�̃�𝑝 : 𝑝 ∈ 𝑓
−1(�̃�)}. By theorem
2.14(i), 𝑓 −1(�̃�) is IS*O in X. Thus 𝑓 −1(�̃�) = 𝑓 −1(Y - �̃�) = X - 𝑓 −1(�̃�) is IS*C in X.
Hence 𝑓 −1(�̃�) is IS*C in X for every ICS �̃� in Y.
(iii)⟹ (iv). Let �̃� be an IS of X and let �̃� be an ICS containing f(�̃�). Then by (iii),
𝑓 −1(�̃�) is IS*C containing �̃�. This implies that IS*cl(�̃�) ⊆ 𝑓 −1(𝑈) and hence
f(IS*cl(�̃�)) ⊆ �̃�. Thus f(IS*cl(�̃�)) ⊆ Icl(f(�̃�)).
(iv)⟹ (v).Let �̃� be an IS of Y. Let �̃� = 𝑓 −1(�̃�). By assumption, f(IS*cl(�̃�)) ⊆ Icl(f(�̃�) ⊆
Icl(�̃�).This implies (IS*cl(�̃�)) ⊆ 𝑓 −1(Icl(�̃�)). Hence IS*cl(𝑓 −1(�̃�)) ⊆ 𝑓 −1(Icl(�̃�)).
(v)⟹ (vi). Let �̃� be an ICS in Y. Then by (v), IS*cl(𝑓 −1(�̃�)) ⊆ 𝑓 −1(Icl(�̃�)) = 𝑓 −1 (�̃�).
Also we have 𝑓 −1 (�̃�) ⊆ IS*cl(𝑓 −1(�̃�)). Hence IS*cl(𝑓 −1(�̃�)) = 𝑓 −1 (�̃�). Thus by
theorem 2.15(ii), 𝑓 −1 (�̃�) is closed. Therefore by lemma 3.11 (ii) Iint*(Icl(𝑓 −1(�̃�))) =
Iint*(𝑓 −1 (�̃�)).
(vi)⟹ (vii). Let �̃� be an IOS in Y. Then Y - �̃� is ICS in Y. Therefore by assumption,
Iint*( Icl(𝑓 −1(Y - �̃�))) = Iint*(𝑓 −1 (Y - �̃�)).This implies that Icl*(Iint (𝑓 −1(�̃�))) =
Icl*(𝑓 −1 (�̃�)).
(vii)⟹ (i). Let �̃� be an IOS in Y. Then by assumption, Icl*(Iint (𝑓 −1(�̃�))) = Icl*(𝑓 −1
(�̃�)). Now by lemma 3.11 (i), 𝑓 −1 (�̃�) is IS*O in X. Hence f is intuitionistic semi *
continuous.
(i)⟹ (viii). Let �̃� be any IS of Y. Then Iint(�̃�) is IOS in Y. By intuitionistic semi *
continuity of f, 𝑓 −1(Iint(�̃�)) is IS*O in X and it is contained in 𝑓 −1 (�̃�). Hence
𝑓 −1(Iint(�̃�)) ⊆ IS*int(𝑓 −1(�̃�)).
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On Intuitionistic Semi * Continuous Functions
(viii)⟹ (i). Let �̃� be an IOS in Y. Then Iint(�̃�) = �̃�. By (viii) 𝑓 −1(�̃�) ⊆ IS*int(𝑓 −1(�̃�))
and hence 𝑓 −1(�̃�) = IS*int(𝑓 −1(�̃�)). Therefore by theorem 2.15(i), 𝑓 −1(�̃�) is IS*O in
X. Thus f is intuitionistic semi * continuous.
Theorem 3.13 The function f: X → Y is not an intuitionistic semi * continuous at an IP 𝑝
in X if and only if 𝑝 belongs to the intuitionistic semi * frontier of the inverse image of
some IOS in Y containing f (𝑝).
Proof: Let f be not an intuitionistic semi * continuous at an IP 𝑝. Then there is an IOS
�̃� in Y containing f(𝑝) such that f(�̃�) is not an IS of �̃� for every IS*O set �̃� in X
containing 𝑝. Hence �̃� ∩ (X - 𝑓 −1(�̃�)) ≠ ∅̃𝐼 for every IS*O set �̃� containing 𝑝. By
theorem 2.16, 𝑝 ̃ ∈ IS*cl(X - 𝑓 −1(�̃�)). Also we have 𝑝 ̃ ∈ 𝑓 −1(�̃�) ⊆ IS*cl (𝑓 −1(�̃�)).
Thus 𝑝 ̃ ∈ IS*cl (𝑓 −1(�̃�)) ∩ IS*cl(X - 𝑓 −1(�̃�)). Hence by theorem 2.18, 𝑝 ∈
IS*Fr(𝑓 −1(�̃�)). Conversely, let f be an intuitionistic semi * continuous at an IP 𝑝. Let �̃�
be any IOS in Y containing f(�̃�). Then 𝑓 −1(�̃�) is an IS*O set in X containing 𝑝. Hence
by theorem 2,15(i), 𝑝 ∈ IS*int(𝑓 −1(�̃�)). Thus �̃� ∉ IS*Fr(𝑓 −1(�̃�)). This proves the
theorem.
Theorem 3.14 Let f: X → ∏ 𝑋𝛼 be an intuitionistic semi * continuous where ∏ 𝑋𝛼 is
the intuitionistic product topology and f(𝑝)= (fα(𝑝)). Then each coordinate function fα : X
→ Xα is an intuitionistic semi * continuous.
Proof: Let �̃� be an IOS in Xα. Then 𝑓𝛼
−1(�̃�) = (Pα ° f)
-1(�̃�) = 𝑓 −1(𝑃𝛼
−1 (𝑈)), where Pα :
∏ 𝑋𝛼 → X, the projection map. Since Pα is intuitionistic continuous, 𝑃𝛼
−1 (�̃�) is IOS in
∏ 𝑋𝛼. Since f is intuitionistic semi * continuous, 𝑓𝛼
−1(�̃�) = 𝑓 −1(𝑃𝛼
−1 (�̃�)) is IS*O in X.
Thus each fα is intuitionistic semi * continuous.
Remark 3.15 The converse of the above theorem is not true in general. However the
converse is true if IS*O(X) is ICS under finite intersection as seen in the following
theorem.
Theorem 3.16 Let f: X → ∏ 𝑋𝛼 be defined by f(𝑝)= (fα(𝑝)) and ∏ 𝑋𝛼 be the
intuitionistic product topology. Let IS*O(X) be ICS under finite intersection. If each
coordinate function fα : X → Xα is intuitionistic semi * continuous, then f is intuitionistic
semi * continuous.
Proof: Let �̃� be the basic IOS in ∏ 𝑋𝛼. Then �̃� = ∩ 𝑃𝛼
−1(�̃�) where each �̃� is IOS in Xα,
the intersection being taken over finitely many α’s and where Pα : ∏ 𝑋𝛼 → X is the
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G. Esther Rathinakani and M. Navaneethakrishnan
projection map. Now 𝑓 −1(�̃�) = 𝑓 −1(∩(𝑃𝛼
−1(�̃�𝛼))) = ∩ 𝑓
−1 (𝑃𝛼
−1(�̃�𝛼)) = ∩ (Pα ° f)
-
1(�̃�𝛼) = ∩ 𝑓𝛼
−1(�̃�𝛼) is IS*O, by hypothesis. Thus by theorem 3.7, f is intuitionistic semi
* continuous.
Theorem 3.17 Let f: X → Y be intuitionistic continuous and g: X → Z be intuitionistic
semi * continuous. Let h: X → Y × Z be defined by h(𝑝) = (f(𝑝), g(𝑝)) and Y × Z be the
intuitionistic product topology. Then h is intuitionistic semi * continuous.
Proof: Using theorem 3.7, it is sufficient to prove that the inverse image under h of
every basic IOS in Y × Z is IS*O in X. Let �̃� × �̃� be the basic IOS in Y × Z. Then ℎ−1
(�̃� × �̃�) = 𝑓 −1 (�̃�) ∩ 𝑔−1 (�̃�). Now by intuitionistic continuity of f, 𝑓 −1 (�̃�) is IO in X
and by intuitionistic semi * continuity of g, 𝑔−1 (�̃�) is IS*O in X. Therefore by theorem
2.14(ii), ℎ−1 (�̃� × �̃�) = 𝑓 −1 (�̃�) ∩ 𝑔−1 (�̃�) is IS*O in X. Hence h is intuitionistic semi
* continuous.
Theorem 3.18 Let f: X → Y be intuitionistic semi * continuous and h: Y → Z be an
intuitionistic continuous. Then h ° f: X → Z is intuitionistic semi * continuous.
Proof: Let �̃� be an IOS in Z. Since h is intuitionistic continuous, ℎ−1 (𝑈) is IOS in Y.
Since f is intuitionistic semi * continuous, 𝑓 −1 (ℎ−1 (�̃�)) is IOS in X. Therefore 𝑓 −1
(ℎ−1 (�̃�)) = (ℎ ° 𝑓)−1) (�̃�) is IS*O in X. Hence h ° f is intuitionistic semi * continuous.
Remark 3.19 From the above theorem it can be seen that the composition of two
intuitionistic semi * continuous need not be intuitionistic semi * continuous.
Example 3.20 Let X = Y = Z ={i, j, k} and τ1= {𝑋�̃�, ∅̃𝐼 , < X, {i},{j, k} >, < X, {k}, {i, j}
>, < X, {i, k}, {b}>}, τ2= {𝑋�̃�, ∅̃𝐼 , < X, {j},{i, k} >, < X, {i},{j} >, < X, {i, j}, ∅>} τ3=
{𝑋�̃�, ∅̃𝐼 , < X, {j},{i} >, < X, {i, j}, ∅>}. Let f: (X, τ1) → (Y, τ2) be defined by f(i) = i, f(j)
= k, f(k) = j and Let g: (Y, τ2) → (Z, τ3) be defined by g(i) = j, g(j) = i, g(k) = k. Then f
and g are intuitionistic semi * continuous. Let g ° f : (X, τ1) → (Z, τ3) and �̃� = < X, {j},
{i}>. Then (𝑔 ° 𝑓 )−1(�̃�) = g(f(< X, {j}, {i}>)) = g(<X, {k},{i}>) = <X, {k},{i}> is not
an IS*O in (Z, τ3). Therefore g ° f is not an intuitionistic semi * continuous.
Definition 3.21 A function f: X → Y is said to be intuitionistic contra semi * continuous
if 𝑓 −1 (�̃�) is IS*C in X for every IOS �̃� in Y.
Remark 3.22 The concept of intuitionistic semi * continuity is free from intuitionistic
contra semi * continuity.
Theorem 3.23 Let f: X → Y be the function. Then the following are equivalent
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On Intuitionistic Semi * Continuous Functions
(i) f is intuitionistic contra semi * continuous.
(ii) For each 𝑝 ̃ ∈ X and each ISC �̃� in Y containing f(𝑝), there exists a IS*O set �̃�
in X containing 𝑝 such that f(�̃�) ⊆ �̃�.
(iii) The inverse image of each ISC in Y is IS*O in X.
(iv) Icl*(Iint(𝑓 −1 (�̃�))) = Icl*(𝑓 −1 (�̃�)) for every ICS �̃� in Y.
(v) Iint*(Icl(𝑓 −1 (�̃�))) = Iint*(𝑓 −1 (�̃�)) for every IOS �̃� in X.
Proof: (i) ⇒ (ii). Let f: X → Y be the intuitionistic contra semi * continuous function.
Let 𝑝 ∈ X and �̃� be an ICS in Y containing f(𝑝). Take �̃� = Y - �̃�. Then �̃� is an IOS in Y
not containing f(𝑝). Since f is intuitionistic contra semi * continuous, 𝑓 −1 (�̃�) is a
intuitionistic semi * closed set in X not containing 𝑝. Therefore 𝑓 −1 (�̃�) = X - 𝑓 −1 (�̃�) is
a IS*C in X not containing 𝑝. Thus �̃� = 𝑓 −1 (�̃�) is a IS*O in X containing 𝑝 such that
f(�̃�) ⊆ �̃�. Hence (i).
(ii) ⇒ (iii). Let �̃� be an ICS in Y and 𝑝 ∈ 𝑓 −1 (�̃�). Then f(𝑝) ∈ �̃�. By assumption, there
is an IS*O set �̃�𝑝 in X containing 𝑝 such that f(𝑝) ∈ f(�̃�𝑝) ⊆ �̃�. Therefore �̃�𝑝 ⊆ 𝑓
−1 (�̃�).
Thus 𝑓 −1 (�̃�) = ∪ { �̃�𝑝 : 𝑝 ∈ 𝑓
−1 (�̃�)}. By theorem 2.14 (i) 𝑓 −1 (�̃�) is an IS*O in X.
Hence (ii).
(iii) ⇒ (iv). Let �̃� be an ISC in Y. Then by hypothesis, 𝑓 −1 (�̃�) is IS*O in X. Hence
from lemma 3.11(i), we have Icl*(Iint(𝑓 −1 (�̃�))) = Icl*(𝑓 −1 (�̃�)).
(iv) ⇒ (v). Let �̃� be an IOS in Y, Then Y - �̃� is an ICS in Y. By assumption,
Icl*(Iint(𝑓 −1 (Y - �̃�))) = Icl*(𝑓 −1 (Y - �̃�)). Therefore [Icl*(Iint(𝑓 −1 (Y - �̃�)))]c =
[Icl*(𝑓 −1 (Y - �̃�))]c. Hence Iint*(Icl(𝑓 −1 (�̃�))) = Iint*(𝑓 −1 (�̃�)).
(v) ⇒ (i). Let �̃� be an IOS in Y. Then by assumption, Iint*(Icl(𝑓 −1 (�̃�))) = Iint*(𝑓 −1
(�̃�)). Therefore by lemma 3.11 (ii), 𝑓 −1 (�̃�) is IS*C in X. Thus f is an intuitionistic
contra semi * continuous.
Theorem 3.24 Every intuitionistic contra continuous is intuitionistic contra semi *
continuous.
Proof: Let f: X → Y be the intuitionistic contra continuous function and 𝑈 be an IOS in
Y. Then 𝑓 −1 (�̃�) is an ICS in X. Hence f is intuitionistic contra semi * continuous.
Theorem 3.25 Every intuitionistic contra semi * continuous is intuitionistic contra semi
continuous.
Proof: Let f: X → Y be the intuitionistic contra semi * continuous function and �̃� be an
IOS in Y. Then 𝑓 −1 (�̃�) is an IS*C in X. Hence f is intuitionistic contra semi continuous.
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G. Esther Rathinakani and M. Navaneethakrishnan
Theorem 3.26 Let (X, τ1) and (Y, τ2) be an ITS.
(i) If f: X → Y is intuitionistic contra semi * continuous and g: Y → Z is
intuitionistic contra continuous, then g ° f: X → Z is intuitionistic semi * continuous.
(ii) If f: X → Y is intuitionistic semi * continuous and g: Y → Z is intuitionistic
contra continuous, then g ° f: X → Z is intuitionistic contra semi * continuous.
(iii) If f: X → Y is intuitionistic contra semi * continuous and g: Y → Z is
intuitionistic continuous, then g ° f: X → Z is intuitionistic contra semi * continuous.
Proof: Let �̃� be an IOS in Z. Then 𝑔−1 (�̃�) is an ICS in Y. Since f is intuitionistic contra
semi * continuous, (g ∘ f)¯1(�̃�) = 𝑓 −1 (𝑔−1 (�̃�)) is IS*C in X. Thus g ° f is intuitionistic
semi * continuous.
(ii) and (iii) can be proved in a similar way.
4. Conclusions
In this paper, we dealt with intuitionistic semi * continuous and intuitionistic contra
semi * continuous. In future we wish to do our research work in intuitionistic semi *
separated, intuitionistic semi * connected, intuitionistic semi * compact, intuitionistic
semi * irresolute continuous function and so on.
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