In the work using interval mathematics, we develop knowledge for cardinal numbers from the viewpoint of uncertainty analysis. In the finite non-T 1 topological spaces, the inclusion-exclusion formula provide interval estimations for the closure and interior of given sets. This paper introduces a novel approach that combines combinatorial and point-set topology, which leads to a number of results. Among these is the cardinality estimation for the intersection of two open sets that cover a hyperconnected topo-logical space

Dochviri, I,

Peters, J,

English

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ON INTERVAL UNCERTAINTIES OF CARDINAL
NUMBERS OF SUBSETS OF FINITE SPACES WITH
TOPOLOGIES WEAKER THAN T 1
J Peters, I Dochviri
To cite this version:
J Peters, I Dochviri. ON INTERVAL UNCERTAINTIES OF CARDINAL NUMBERS OF SUBSETS
OF FINITE SPACES WITH TOPOLOGIES WEAKER THAN T 1. 2020. ￿hal-02464752￿
ON INTERVAL UNCERTAINTIES OF CARDINAL
NUMBERS OF SUBSETS OF FINITE SPACES WITH
TOPOLOGIES WEAKER THAN T1
J.F. PETERS AND I.J. DOCHVIRI
Dedicated to A. V. Arkhangel’skii and S.A. Naimpally
Abstract. In the work using interval mathematics, we develop
knowledge for cardinal numbers from the viewpoint of uncertainty
analysis. In the finite non-T1 topological spaces, the inclusion-
exclusion formula provide interval estimations for the closure and
interior of given sets. This paper introduces a novel approach that
combines combinatorial and point-set topology, which leads to a
number of results. Among these is the cardinality estimation for
the intersection of two open sets that cover a hyperconnected topo-
logical space.
Contents
1. Introduction 2
2. Preliminaries 2
3. Main Results 4
References 6
2010 Mathematics Subject Classification. Primary 54A25, Secondary 05A19.
Key words and phrases. Finite sets, cardinal number, hyperconnected, inclusion-
exclusion property, interval mathematics, topology.
The research has been supported by the Natural Sciences & Engineering Research
Council of Canada (NSERC) discovery grant 185986 and Instituto Nazionale di Alta
Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture Algebriche,
Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U 2016/000036 and
Scientific and Technological Research Council of Turkey (TÜBİTAK) Scientific Human
Resources Development (BIDEB) under grant no: 2221-1059B211301223.
1
2 J.F. PETERS AND I.J. DOCHVIRI
1. Introduction
In point-set topology, trends of last five decades are connected with
investigations of infinite cardinal functions of topological spaces. How-
ever, many combinatorial properties of finite topological spaces are in the
shadow. Even so, discrete mathematics and combinatorics use finite sets
for naturally formulated inclusion-exclusion identities via closure opera-
tors [4] and improved Bonferroni inequalities via abstract tubes [5], [8].
Also, we must emphasize, that characterizations of finite sets yield im-
portant results in computational topology (see, e.g., [9], [2]).
It seems that cardinality counting problems in discrete mathematics
and combinatorics began using famous inclusion-exclusion formula. That
is, for two given finite sets, we have the well-known view, namely,
card(A ∪B) = card(A) + card(B)− card(A ∩B).
This formula is successfully applicable, if we know exact values of cardinal-
ities (see, e.g., [8], [4], [3]). However, it is interesting that the evaluation
of cardinalities of corresponding sets occurs while we have imprecise infor-
mation about cardinals of the particular sets. Such a situation arises when
we have, for example, so-called big data sets and molecular structures.
2. Preliminaries
In [7], R. Moore developed interval mathematics for computational
problems, where parameters of investigating models are uncertain and we
are only able to describe parameters by closed interval estimations. In
this section, we briefly recall basic operations of interval arithmetic.
Let a1, a2, b1, b2, x ∈ R. A closed interval of the reals is denoted by
[a1, a2] = {x ∈ R : a1 ≤ x ≤ a2}. From [7], we have following interval
arithmetic:
(1) [a1, a2] + [b1, b2] = [a1 + b1, a2 + b2];
(2) [a1, a2]− [b1, b2] = [a1 − b2, a2 − b1];
(3) [a1, a2]×[b1, b2] = [minP,maxP ], where P = {a1b1, a1b2, a2b1, a2b2};
(4) If 0 /∈ [b1, b2], then [a1,a2][b1,b2] = [a1, a2]× [ 1b2 , 1b1 ].
It should be especially noticed that any real number k is identified
with interval [k, k]. Moreover, if a1 and b1 are non-negative real numbers
then interval multiplication (3) should be change in the following way
[a1, a2] × [b1, b2] = [a1b1, a2b2]. Also, if a1 ≥ 0 and b1 > 0 then the rule
(4) can be simplified as following [a1,a2][b1,b2] = [
a1
b2
, a2b1 ].
SUBSET CARDINALITY IN FINITE TOPOLOGICAL SPACES WEAKER THAN T13
There are established several important computational differences be-
tween interval arithmetic from real one, but we do not need more infor-
mation than we present here about interval mathematics.
In the sequel, the sets of natural and rational numbers are denoted by
symbols N and Q, but N0 = N ∪ {0}.
For a rational number q ∈ Q we have to use two well-known notations:
⌊q⌋ = max{m ∈ Z|m ≤ q} and ⌈q⌉ = min{n ∈ Z|n ≥ q}.
For topological spaces we use notions from [1]. If O ⊂ X is an open
subset of a topological space (X, τ) then we will write O ∈ τ . The comple-
ment of an open set is called to be closed set. The collection of all closed
subsets of (X, τ) we denote by co(τ). Also, in a finite topological space
(X, τ) denote by cl(A) closure (resp. int(A) interior of) A ⊂ X, which is
minimal closed (resp. maximal open) set containing (resp. contained in)
a set A.
Recall that a topological space (X, τ) is T1 space if and only if {x}
is closed set, for every x ∈ X. Therefore, in T1 space (X, τ), we have
{x} = cl({x}) for every x ∈ X.
For the finite, non-T1 topological spaces cardinal estimations using clo-
sure and interior operators is less lightened part of discrete mathematics.
Naturally, if we know about a set A that both of estimations card(A) ∈
[a1, a2] and card(A) ∈ [b1, b2] are valid, where [a1, a2]∩ [b1, b2] ̸= {∅} then
we should declare card(A) ∈ [max{a1, b1},min{a2, b2}].
Theorem 2.1. Let A ⊂ B be subsets of a set X where card(X) = n and
card(A) ∈ [a1, a2]. Then card(B) ∈ [a1 + 1, n− 1].
Proof. It is obvious that A ⊂ B implies that card(A) < card(B). Since
the minimal value of cardinality of a set A can be equal to a1, then
a1 + 1 ≤ card(B). On the other hand we have, B ⊂ X and card(B) <
card(X) = n. Hence it can be write card(B) ≤ n− 1.

Theorem 2.2. Let A, B and C be finite subsets of a set X such that
C = A × B, card(C) ∈ [c1, c2] and card(A) ∈ [a1, a2]. Then card(B) ∈
[⌈ c1a2 ⌉, ⌊ c2a1 ⌋].
Proof. Since for Cartesian product C = A×B, we can write following car-
dinal equality: card(B) = card(C)card(A) , then applying above mentioned opera-
tion of the interval division we get card(B) ∈ [ c1a2 , c2a1 ]∩N0 = [⌈ c1a2 ⌉, ⌊ c2a1 ⌋].

4 J.F. PETERS AND I.J. DOCHVIRI
Theorem 2.3. Let X = A ∪B be a finite set with card(X) ∈ [m,n], but
card(A) ∈ [a1, a2] and card(B) ∈ [b1, b2]. Then card(A ∩B) ∈ [a1 + b1 −
n, a2 + b2 −m] ∩ N0.
Proof. Applying the famous inclusion-exclusion formula, we can write
card(A∩B) = card(A) + card(B)− card(A∪B) = card(A) + card(B)−
[m,n]. By substitution of given cardinal estimations we obtain card(A ∩
B) ∈ [a1 + b1 − n, a2 + b2 −m] ∩ N0. 
3. Main Results
In section, we work with topological spaces which are not even T1
topologies. Examples of such topological spaces are known in the point-
set topology as T0 and R0 spaces.
In the Theorem 3.1, Theorem 3.2 and Theorem 3.3, we assume that
(X, τ) is a non-T1 space with card(X) = n and min{card(F )|F ∈ co(τ) \
{∅}} = 2.
Theorem 3.1. Let A = {a1, a2, ..., am} be a subset of a topological space
(X, τ) with m ∈ [1, ⌊n2 ⌋]. If cl({ai}) ∩ cl({aj}) = ∅, for i, j ∈ {1, 2, ...,m}
and i ̸= j then card(cl(A)) ∈ [2m,n].
Proof. It is known that in T1 topological space (X, τ) with card(X) = n
we have card(cl({x})) = 1, for every x ∈ X. Therefore, in view our
conditions we conclude that n ≥ card(cl({x})) ≥ 2, for every x ∈ X. Note
that for the set A = {a1, a2, ..., am} we can write its closure as following:
cl(A) = cl({a1})∪cl({a2})∪...∪cl({am}). Therefore, the inequalities hold
n ≥ card(cl(A)) = card(cl({a1}))+card(cl({a2}))+...+card(cl({am})) ≥
2m. 
Theorem 3.2. Let A = {x1, x2, ..., xp} be a subset of a topological space
(X, τ) with p ∈ [⌈n2 ⌉, n]. If card(cl(x)) ∈ [2, kx], for any point x ∈ (X \A)
and cl({xi}) ∩ cl({xj}) = ∅, for i, j ∈ {p+ 1, p+ 2, ..., n}, i ̸= j, then
card(int(A)) ∈ [0, 2p− n], if n ≤∑x∈(X\A) kx
and
card(int(A)) ∈ [n−∑x∈(X\A) kx, 2p− n], if ∑x∈(X\A) kx < n.
Proof. Assume that X = {x1, x2, ..., xn} and A = {x1, x2, ..., xp}. It is
known that int(A) = X \ cl(X \A). Since card(X \A) = n−p then using
Theorem 3.1. we can write card(cl(X \ A)) ∈ [2(n − p), n]. But, taking
into account condition card(cl(xi)) ∈ [2, ki], i = p+ 1, n we obtain better
estimation than previous, namely: card(cl(X \ A)) = card(cl(xp+1)) +
SUBSET CARDINALITY IN FINITE TOPOLOGICAL SPACES WEAKER THAN T15
card(cl(xp+2)) + ... + card(cl(xn)) ∈ [2, kp+1] + [2, kp+2] + ... + [2, kn] =
[2(n− p),min{n,∑ni=p+1 ki}].
It is clear that if n ≤ ∑ni=p+1 ki then card(cl(X \ A)) ∈ [2(n −
p), n]. Hence we get card(int(A)) ∈ [0, 2p − n]. If ∑ni=p+1 ki < n then
card(cl(X \ A)) ∈ [2(n − p),∑ni=p+1 ki] and we obtain card(int(A)) ∈
[n−∑ni=p+1 ki, 2p− n]. 
Recall that a set A of a topological space (X, τ) is called semi-open if
there exists O ∈ τ\{∅} such that O ⊂ A ⊂ cl(O) [6]. The complement of
an semi-open set is called semi-closed. The class of all semi-open (resp.
semi-closed) subsets of a space (X, τ) we denote usually as SO(X) (resp.
SC(X)). It can be easily to verify that A ∈ SO(X) if and only if A ⊂
cl(intA), but B ∈ SC(X) if and only if int(clB) ⊂ B.
Theorem 3.3. Let A ∈ SO(X) be a nonempty subset of (X, τ). Then
there exists k ∈ N such that card(A) ∈ [k + 1, 2k − 1], where k ∈ [1, ⌊n2 ⌋].
Proof. For a set A ∈ SO(X) we can choose O ∈ τ\{∅} such that O ⊂
A ⊂ cl(O). Hence card(O) < card(A) < card(cl(O)) ≤ n. Denote by
k = card(O), then it is obvious that k ∈ [1, n − 1]. Hence card(A) ∈
[k+1, n−1], but by Theorem 3.1. we can write card(clO) ∈ [2k, n]. Note
that the inequality 2k < n implies k ∈ [1, ⌊n2 ⌋]. Collecting our estimations
we get k + 1 ≤ card(A) < [2k, n], i.e. card(A) ∈ [k + 1, 2k − 1]. 
A topological space (X, τ) is called hyperconnected if cl(O) = X, for
every O ∈ τ \ {∅}. It is obvious that (X, τ) is hyperconnected if and only
if O1 ∩O2 ̸= {∅}, for any pair of O1, O2 ∈ τ \ {∅}.
Now, in contrast of above theorems we remove certain conditions from
a topological space.
Theorem 3.4. Let (X, τ) be a hyperconnected topological space with card(X) ∈
[m,n]. If X = O1 ∪ O2, where O1, O2 ∈ τ\{∅} are sets with card(O1) ∈
[a1, a2], card(O2) ∈ [b1, b2] and max{a2, b2} < m. Then card(O1 ∩O2) ∈
[a1 + b1 − n, a2 + b2 −m] ∩ N0.
Proof. Since in the hyperconnected space (X, τ) we have O1 ∩ O2 ̸= ∅,
for any pair of O1, O2 ∈ τ \ {∅} then it takes place following equality:
card(O1∩O2) = card(O1)+card(O2)−card(O1∪O2) = [a1, a2]+[b1, b2]−
[m,n] = [a1 + b1 − n, a2 + b2 −m] ∩ N0.

6 J.F. PETERS AND I.J. DOCHVIRI
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Taimanov, Zbl 1268.54001.
Department of Electrical and Computer Engineering, University of Man-
itoba and Department of Mathematics, Faculty of Arts and Sciences, Adiya-
man University, 02040 Adiyaman, Turkey
Email address: james.peters3@ad.umanitoba.ca
Department of Mathematics; Caucasus International University; 73 Char-
gali str., 0141 Tbilisi, Georgia
Email address: iraklidoch@yahoo.com


ON INTERVAL UNCERTAINTIES OF CARDINAL NUMBERS OF SUBSETS OF FINITE SPACES WITH TOPOLOGIES WEAKER THAN T 1

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ON INTERVAL UNCERTAINTIES OF CARDINAL NUMBERS OF SUBSETS OF FINITE SPACES WITH TOPOLOGIES WEAKER THAN T 1

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