INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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ISSN 2278-2540 | DOI: 10.51583/IJLTEMAS | Volume XIII, Issue X, October 2024
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Lattices of Reduction and Subset-Induced Topologies
Alexander O. Ilo., Chika S. Moore., Precious N. Ugwueze
Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State
DOI : https://doi.org/10.51583//IJLTEMAS.2024.131027
Received: 08 November 2024; Accepted: 18 November 2024; Published: 22 November 2024
Abstract: Induced topologies have been studied only from the standpoint of a superset down to its subset to get what we call
subspace topology. Here we turn the focus around and show that subsets can induce topologies on their supersets. Also, induced
topologies have so far only been constructed by collecting the intersections of open sets of a superset with a subset. Here again we
extend the focus and show that a superset will always induce topologies on their subsets through other means than by taking
intersections of open sets with a subset. All these warrant further research into a more extensive and comprehensive study of
induced topologies; to establish how some topological properties such as compactness, separation axioms, etc. are shared or
inherited in the wider context of inducement of topologies. The concept of reducible topologies has been explored and published
by the authors before [1]. Here we extend the research by proving that any pairwise comparable family F of subsets of a set X
generates a reducible topology τ on X, and that the chain C of reductions of τ can be constructed in such a way that card(F) =
card(C).
Keywords: Induced Topology, Reduction of Topology, Lattice of Reductions, Chain of Reductions, Comparison of Topologies
Mathematics Subjects Classification (MSC) 2020: 54A05, 54A10
Reducible Topologies—Lattices
I. Introduction
We recall the following definitions.
Definition 1.1 A relation R on a set X is called a partial order on X if
1. R is reflexive; in that xRx, for all x X,
2. R is transitive; in that xRy and yRz implies xRz,
3. R is anti-symmetric; in that xRy and yRx implies x = y.
Definition 1.2 A set X on which a partial order is defined is called a partially ordered set; in brief, a poset.
Definition 1.3 If X is a poset, with partial order R, and xRy, then we say that x precedes y, written x y. We then analogously
also say that y dominates x. If x precedes y and x ≠ y, we say that x properly precedes y, or y properly dominates x.
Definition 1.4 Let X be a poset with R. Then x is called a lower bound of y if x y; and then y is called an upper bound of x.
Definition 1.5 Let X be a poset with R. An element x
0
of X is called the first or the least element of X if x
0
precedes every other
element of X. The last or greatest element of X is that which dominates every other element of X.
Definition 1.6 Let X be a poset. An element x
0
of X is called a minimal element if no element of X properly precedes x
0
.
If x
0
is a minimal element of a poset X and x x
0
, then x = x
0
. Also, every first element is a minimal element but a minimal
element may not be a first element.
Definition 1.7 Let X be a poset. An element y
0
of X is called a maximal element if no element of X properly dominates y
0
.
Definition 1.8 Let X be a poset. Let T be a subset of X. A lower bound of T is an element of X which precedes every element of
T. The greatest lower bound (g.l.b.) of T is the lower bound which dominates every other lower bound of T. The g.l.b. of T is also
called the infimum of T, and denoted inf(T).
Definition 1.9 Let X be a poset and let T be a subset of X. An upper bound of T is an element of X which dominates every
element of T. The least upper bound (l.u.b.) of T is the upper bound which precedes every other upper bound of T. The l.u.b. of T
is also called the supremum of T, and denoted sup(T).
Definition 1.10 Two elements x, y of a poset X are said to be comparable if either x y or y x.
Definition 1.11 A lattice is a poset in which every two elements have a g.l.b and an l.u.b.
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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II. Development of Lattice of Topologies
Let C = {τ
α
: α ∆} be a chain of reductions of a topology τ on a set X. Then C, with the relation of set inclusion is a poset. We
also see that C is totally ordered. Ifτ
α
and τ
α
are two topologies in C such that, say, τ
α
is weaker than τ
α
, then the g.l.b. of the
sub-family T = {τ
α
,τ
α
} of C, that is, inf(T), is τ
α
. Also sup(T) = τ
α
. Hence C is a lattice of topologies by set inclusion.
Let R be another relation on the chain C, where τ
α
Rτ
r
if τ
α
≤ τ
r
. That is, the relation R (≤) on C, now, is that of comparison of
topologies. With this relation on C, we see again that C is a lattice of topologies. What we have just established is the following.
Corollary 1.12 Every chain C of reductions of a topology on a set X is a lattice in at least two ways.
Observations
Every set on which a partial order is defined is not a lattice; that is, not every poset that is a lattice. In particular, every family of
topologies is not a lattice. For example, if the topologies in a family F are not comparable, then the family F would not be a lattice
in either of the ways; but F would still be a poset in the two ways.
If a family of subsets of a set X is pairwise comparable by set inclusion (i.e. totally ordered by set inclusion), then it generates a
topology (on X) which has a chain of reductions. This indeed is a theorem which marks the end and climax of this section.
Theorem 1.1 Any (set inclusion) pairwise comparable family F of subsets of a set X generates a reducible topology τ on X. And
the chain C of reductions of τ can be constructed in such a way that card(F) = card(C).
Proof: Let F = {A
α
: A
α
X}
α
∆
be a family of (set inclusion) pairwise comparable subsets of X. Let A
α1
and A
α2
be two elements
of F such that, say, A
α1
A
α2
. Let γ
1
= A
α1
-induced topology on X and γ
2
= A
α2
-induced topology on X. If γ
1
and γ
2
are not
comparable, let τ
1
= γ
1
and τ
2
= γ
1
▽ γ
2
, the join of γ
1
and γ
2
(defined as the weakest topology, on X, finer than both γ
1
and γ
2
). Then
τ
1
and τ
2
are two comparable topologies on X. Precisely, τ
1
is strictly weaker than τ
2
.
Since F is pairwise comparable, the sets in F can be arranged such that
A
α
A
r
···.
It follows from the construction above that these sets in F have, corresponding to them, a family C = {τ
α
}
α
∆
of topologies on X,
which is pairwise comparable in that
τ
α
≤ τ
r
≤ ···.
It is easy to see that C is equivalent to F; that is, card(C) = card(F).
■
It is easier to see the existence of the chain C, constructed in the proof of the theorem if we remember that the construction can
actually be done through inducement by the discrete topologies of A
α1
and A
α2
; or, by what is similar, first getting a topology on
A
α2
and then using this to induce a topology on A
α1
; and then finally using these two topologies to construct subset-induced
topologies on X.
III. Subset-induced Topologies
Proposition 2.1 If X E, then any topology, say τ
X
, on X induces a topology, say τ
XE
, on E, given by τ
X
(E) = τ
X
{E}.
Proof: It is easy to see that τ
X
(E), since τ
X
. Also E τ
X
(E), by definition. Let {G
i
: i = 1, . . ., n} be a sub-collection of
τ
X
(E). We show that the intersection
belongs to τ
X
(E). Clearly
τ
X
(E) if any of the
comes from τ
X
. If all the
are each equal to E, then E =
is an element of τ
X
(E). Hence in any case τ
X
(E) is closed under finite intersections. Let
α
α be any family of sets in τ
X
(E). If one of these sets equals E, then their union would equal E, which belongs to τ
X
(E).
If none of these sets equals E, then each of them belongs to τ
X
and hence their union belongs to τ
X
which is itself a subfamily of
τ
X
(E). These imply that τ
X
(E) is also closed under arbitrary unions, and is therefore a topology on E.
■
Definition 2.1 The topology τ
X
(E), on E, is called an X-topology on E; or a topology induced on E by the topology τ
X
on X.
Observe that one subset can induce several topologies on its superset.
Proposition 2.2 Let (E,τ) be a topological space, and let X τ be a τ-open subset of E. Let τ
X
= {G τ : G X}. Then τ
X
is a
topology on X.
Proof:
1. τ
X
, since τ and X.
2. X τ
X
, since X τ and X X.
INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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3. Let {G
i
}
1≤i≤n
τ
X
be any finite number of sets of τ
X
; and let
be the intersection of these sets. Then clearly N τ, as the intersection of a finite number of sets of τ. Also it is clear that N
X, since it is the intersection of some subsets of X. Hence N τ
X
.
4. Let {G
α
}
α
∆
τ
X
be any family of sets of τ
X
. Then G
α
= U τ,
α∆
since τ is closed under arbitrary unions. Also U X, as a union of subsets of X. Hence U τ
X
, implying that τ
X
is closed under
arbitrary unions and, hence, a topology on X.
■
Definition 2.2 With X, E and τ
X
as given in proposition 2.2, let τ
X
(E) = τ
X
{E} be an X-topology on E. Then τ
X
(E) is an open
subset induced topology on E.
IV. Discussions
1. Proposition 2.2 shows that a superset can induce a topology on its subset through other means than by collecting the
intersections of the subset with the open sets of the superset. But someone might still say (against the idea of Proposition
2.2) that since X is open in E, the induced topology τ
X
on X is the same thing as what would have resulted if the
intersections of X with open sets of E were collected. To properly see the difference between the old method and the new
method of inducement here, see proposition 2.3 below; it is the general form of proposition 2.2.
2. Definition 2.2 shows that a subspace topology can in turn induce a topology on its ’superspace’, and that such a subset-
induced topology on a superset may actually be comparable with the original topology of the superset.
Proposition 2.3 Let (E, τ) be a topological space, and let X be any subset of E. Let τ
X
= {G τ : G X}{X}. Then τ
X
is a
topology on X.
Application
Let (R, u) denote the usual topological space of the set of real numbers, and let X = [a, b] be a closed interval in R. Using
proposition 2.3, the u-induced topology on X is the family τ
X
= {G U : G X}{X}. It is clear that no set of the form [a, c) or
(c, b] is open in this induced topology of X, where a < c < b. It is also clear that all such half-open intervals are open in X when
the inducement is done using the old method of collecting the intersections of X with the open sets of (R, u).
V. Summary and Conclusion
1. Induced topologies have been studied only from the standpoint of a superset down to its subset. We showed that subsets
can induce topologies on their supersets.
2. Induced topologies have so far only been constructed by collecting the intersections of open sets of a superset with a
subset [5], [6], [9], [10], and [13]. We expanded the focus and showed that a superset can always induce topologies on
their subsets through other means than by taking intersections of open sets with the subset.
3. Further extensive and comprehensive study of induced topologies needs to be done to establish how some topological
properties such as compactness, separation axioms, etc. are shared in the wider context of inducement of topologies.
4. We proved that any pairwise comparable family F of subsets of a set X generates a reducible topology τ on X, and that the
chain C of reductions of τ can be constructed in such a way that the cardinality of F equals the cardinality of C.
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INTERNATIONAL JOURNAL OF LATEST TECHNOLOGY IN ENGINEERING,
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