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Reducibility of Topologies
Alexander O. Ilo., Chika S. Moore., Chukwunonso Ofodile
Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State, Nigeria
DOI : https://doi.org/10.51583/IJLTEMAS.2024.131007
Received: 20 October 2024; Accepted: 26 October 2024; Published: 05 November 2024
Abstract: The concepts of weak or strong reduction of topologies are introduced. Closely related to these, and introduced as well,
are the concepts of weak and strong base reduction of topologies. We also defined extensible topologies; and defined weak and
strong base extension of topologies. We proved that there exists a topology γ, weaker than a weak topology τ, on X, which has a
chain of strong reductions if one of the range spaces, say (Xα, τα) of τ, has a chain of strong reductions. It is proved that the usual
topology of the set R of real numbers can be reduced in the weak sense to chains of infinite families of pairwise comparable
topologies; and that the usual topology of R can neither be reduced in the normal sense nor in the strong sense. We proved that a
weak topology has a chain of weaker topologies if one of its range topologies is reducible to a chain of topologies.
Keywords: Reduction of Topology, Strong, Normal and Weak Reduction of Topologies, Extension of Topologies, Weak Topology,
Comparability of Topologies, Base Reduction of Topologies Mathematics Subjects Classification (MSC) 2020: 54A05, 54A10
I. Introduction
Throughout, X is a nonempty set.
Definition 1.1 A topology τ on X is said to be strongly reducible or reducible in the strong sense if G τ such that τ1 = τ − {G} is
a topology on X. The topology τ1 is called a strong reduction of τ.
Example 1
Let X = {a, b, c} and τ = {, X, {a}, {c}, {a, c}}. Then τ on X is strongly reducible, since there exists {a} τ such that τ1 = τ −
{{a}} ≡ {, X, {c}, {a, c}} is a topology on X. Conversely, τ1 is a strong reduction of τ.
Let X = {a, b, c} and τ = 2X = {, X, {a},{b},{c},{a ,b},{a, c},{b, c}}. Then τ = 2X is not strongly reducible.
Definition 1.2 A topology τ on X is said to be normally reducible or simply reducible, or reducible in the normal sense if there
exist Gi τ (i = 1,···,m); m IN such that τ1 = τ −{G1,···,Gm} is a topology on X. Such a topology τ1 is called a normal reduction of
τ, or simply a reduction of τ.
Example 2
Let X = {a, b, c} and τ = 2X = {, X,{a},{b},{c},{a, b},{a, c},{b, c}}. Then τ = 2X is normally reducible, to τ1 = τ − {{c},{b, c}} ≡
{, X,{a},{b},{a, b},{a, c}}.
Definition 1.3 A topology τ on X is said to be weakly reducible or reducible in a weak sense if there exist{Gα τ : α ∆} such that
τ1 = τ − {Gα τ : α ∆} is a topology on X. The topology τ1 is called a weak reduction of τ.
Example 3
Let (IR, U) denote set R of real numbers with its usual topology U. Let X = (−∞,0), and τX = {G U : G X}{IR}. Then τX is a
weak reduction of U, since τX = U − {G U : G is not a subset of X}.
Remark
1. Strongly Reducible = Normally Reducible = Weakly Reducible. But the converses are not always true.
2. The indiscrete topology of a set cannot be reduced in any sense (strong, normal or weak). In fact, it is the weakest reduction
of any topology.
3. In the first two examples above we saw that the discrete topology of X is not reducible in the strong sense. This is actually
a general fact for the discrete topology of any set X whose cardinality is greater than 2; and we state and prove that below
as a theorem.
4. The discrete topology is not the only topology that is irreducible in the strong sense. The usual topology of IR is not
reducible in the strong sense. This is stated and proved below as a proposition.
Theorem 1.1 (a) The discrete topology of X cannot be reduced in the strong sense if the cardinality of X is greater than 2. (b) Every
non-indiscrete topology on a set X can be reduced in some sense (strong, normal or weak).
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Proof:
(a) Let the cardinality of X be greater than 2 and let (X, D) be a discrete topological space. Suppose G D and η = D − {G}.
We need to show that η is not a topology on X.
Without loss of generality, suppose G ≠ {a}. Then there exist at least two proper subsets of G and each is in D (as the discrete
topology) and hence separately in η. Since G is the union of all the proper subsets of G, it follows (as G η) that η is not closed
under arbitrary unions and is hence not a topology on X.
Now suppose G = {a}, a singleton. Then from hypothesis X contains two other mutually distinct elements x1, x2, each different from
a. The sets G1 = {a, x1} and G2 = {a, x2} are in D (as the discrete topology) and hence in η. It is easy to see that G1 ∩ G2 = G η;
hence η is not a topology on X.
(b) Let τ be a non-indiscrete topology on X. Then the indiscrete topology {, X} on X is a reduction of τ in some sense. The
proof is complete.
■
Proposition 1.1 The usual topology U of the set R of real numbers is not reducible in the strong sense.
Proof:
Let (R,U) denote IR with its usual topology. Let η = U − {(a ,b)},for some
(a, b) U. We show that η is not a topology on IR. For each n N let
.
Then each Gn is an element of U and an element of η. Clearly
and since (a, b) η it follows that η is not closed under
arbitrary unions and is hence not a topology on IR.
■
Note
Not only that the usual topology of R cannot be reduced in the strong sense; it can also not be reduced in the ’normal’
sense.
There can be found many other topologies which are not reducible in the strong sense. For example the lower limit topology
of R is not strongly reducible and the upper limit topology of R is not strongly reducible. Yet infinitely many topologies
can be reduced in the strong sense—for example, the discrete topology of any set with only two elements has a chain of
strong reductions.
So far, it may appear that the only examples of strongly reducible topologies available are finite topologies or topologies
on finite sets. Infinite topologies and indeed topologies on infinite sets can be strongly reducible. The next example
illustrates this.
Example 4
Let N= {0,1,2,···} denote the set of natural numbers. For each n N let Gn be the set of all real numbers excluding the first n natural
numbers. Thus for instance
G0 = R− {} = R;
G1 = R− {0};
G2 = R− {0, 1};
G3 = R− {0, 1, 2};
.
.
.
Gn = R − {0, 1, 2, 3,···, n − 1}
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Let TCN = {, Gn}n N. Then it is easy to see that
1. The empty set is in TCN, from the way TCN is defined.
2. The whole set R of real numbers is in TCN.
3. TCIN is closed under finite intersections.
4. And that TCIN is closed under arbitrary unions.
Hence TCIN is a topology on IR. We see that TCIN is strongly reducible since, say τ = TCIN− {G5} is a topology on IR. (The topology
TCIN here is one of our interesting constructions in this work.)
Definition 1.4 A topology τ on X, with base B, is said to be strongly base reducible or base reducible in the strong sense if there
exists B0 B such that B1 = B − {B0} is a base for a topology τ1 on X strictly coarser than τ. Such a topology τ1 is called a strong
base reduction of τ.
Example 5
Let X = {a ,b, c} and τ1 on X be τ1 = 2X = {,X,{a},{b},{c},{a, b},{a, c},{b, c}}.
Let B1 = {{a},{b},{c}} be a base for the topology τ1 on X. Then τ1 with the base B1 is not strongly base reducible.
However, if we endow X with the topology τ2 = {,X,{a},{b},{a, b},{a, c}}, with base B2 = {{a},{b},{a, c}}, then τ2 would be
strongly base reducible, for there exists {a} B2 such that B3 = B2 − {{a}} ≡ {{b},{a, c}} is a base for a topology τ3 on X given by
τ3 = {, X,{b},{a, c}}.
Definition 1.5 A topology τ on X, with base B, is said to be base reducible if there exists Bi B(i = 1,···,m; m N) such that B1 =
B−{Bi : i = 1,···,m} is a base for a topology τ1 on X strictly coarser than τ. Such a topology τ1 is called a base reduction of τ.
Definition 1.6 A topology τ on X, with base B, is said to be weakly base reducible or base reducible in the weak sense if there
exist{Bα B : α ∆} such that B1 = B − {Bα : α ∆} is a base for a topology τ1 on X strictly coarser than τ. Such a topology τ1 is
called a weak base reduction of τ.
Example 6
Let (R, U) denote the usual topological space of R. Then B = {(a, b): a, b R} is a base for U. Let B1 = {Bα B: Bα (−∞, 0)}{R}.
Then B1 is a base for a topology on IR (namely the topology τX = {G U: G X}{R} given after Definition 1.3) strictly weaker
than U. That is, the topology τX is a weak base reduction of (R, U).
Remark
A strongly base reducible topology is base reducible. A base reducible topology is weakly base reducible but converses of these do
not hold in general.
Definition 1.7 A topology τ on X is said to be
1. strongly extensible if G X, G / τ such that γ = τ {G} is a topology on X. The topology γ is then called a strong
extension of τ;
2. extensible if {Gi X : Gi τ; i = 1,···,m; m N} such that γ = τ {G1,···,Gm} is a topology on X. The topology γ is called
an extension of τ;
3. weakly extensible if {Gα X : Gα τ;α ∆} such that γ = τ {Gα}α∆ is a topology on X. Such a γ is then called a weak
extension of τ.
Definition 1.8 A topology τ on X with base B is said to be
1. strongly base extensible if B0 X, B0 B such that Ω = B {B0} is a base for a topology γ on X finer than τ. The topology
γ is then called a strong base extension of τ;
2. base extensible if {Bi X, Bi B, i = 1,···,m; m N} such that Ω = B {Bi; i = 1,···,m} is a base for a topology γ on X,
finer than τ. The topology γ is called a base extension of τ;
3. weakly base extensible if {Bα X: Bα B,α ∆} such that Ω = B {Bα : α ∆} is a base for a topology γ on X finer than
τ. In this case the topology γ is called a weak base extension of τ.
The following propositions hold true obviously from the definitions above.
Proposition 1.2 A topology τ on X is
1. strongly extensible if, and only if, τ is a strong reduction of some topology γ on X;
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2. extensible if, and only if, τ is a reduction of some topology γ on X;
3. weakly extensible if, and only if, τ is a weak reduction of some topology γ on X.
Proposition 1.3 A topology τ on X with base B is
1. strongly base extensible if, and only if, τ is a strong base reduction of some other topology γ on X;
2. base extensible if, and only if, τ is a base reduction of some topology γ on X;
3. weakly base extensible if, and only if, τ is a weak base reduction of some topology γ on X.
Definition 1.9 Let τ be a strongly reducible topology on X. If τ1 is a strong reduction of τ, τ2 a strong reduction of τ1, τ3 a strong
reduction of τ2, and so on, then the pairwise comparable family
C = {τn} n N
of topologies on X is called a chain of strong reductions of τ on X.
Example 7
Let X = {a, b, c} and τ on X be τ = {, X, {a}, {c}, {a, c}}. Then τ1 = {, X, {c},{a, c}} or τ1 = {, X, {a}, {a, c}} is a strong
reduction of τ. Also τ2 = {, X, {c}} or τ2 = {, X, {a}} or {, X, {a, c}} is a strong reduction of τ1.
And τ3 = {, X} is a strong reduction of τ2. Hence the family
C1 = {τ1, τ2, τ3}
is a chain of strong reductions of τ.
For the topology τ on X given by τ = {, X, {a}, {c}, {a, c}, {b, c}} a chain of strong reductions can be obtained as follows: τ1 =
{, X, {a}, {c}, {a, c}}; τ2 = {, X,{a},{a, c}}; τ3 = {, X,{a}}; and τ4 = {, X}.
We see that
τ4 < τ3 < τ2 < τ1 < τ;
and that
C2 = {τ1, τ2, τ3, τ4}
is a chain of strong reductions of τ.
Remark
We notice first that a strongly reducible topology can be reduced to a chain of pair-wise comparable topologies. Secondly, there is
often more than one way of getting a chain of strong reductions of a strongly reducible topology.
The chains C1 and C2 in the last example are simple enough, in that they are (each) finite. Hence one may wonder if the only
examples of chain of strong reductions (of a topology) that could be found are those that are finite. Actually examples of
denumerable chains of reductions exist. For example, the topology TCN on R that we constructed above, just before definition 3.6,
has a countably infinite chain of strong reductions. To see this, we observe that
where τ0 = {,IR}, τ1 = τ0 {G1}, τ2 = τ1 {G2}, and so on. Then
C = {τ0, τ1, τ2,···}
is a countably infinite family of strong reductions of TCIN.
Definition 1.10 Let τ be a (strongly or weakly) reducible topology on X. If C1 and C2 are two chains of (weak or strong) reductions
of τ such that for each τ1i C1, there exists τ2j C2 such that τ1i is weaker than τ2j, then we say that the chain C1 is weaker than the
chain C2.
Definition 1.11 Let τ be a (strongly or weakly) reducible topology on X. If C1 and C2 are two chains of (weak or strong) reductions
of τ such that for each τ1i C1, there exists a τ2j C2 such that τ1i is strictly weaker than τ2j, then we say that the chain C1 is strictly
weaker than the chain C2.
Definition 1.12 If C1 and C2 are two chains of reductions of τ on X such that C1 is weaker than C2 and C2 is weaker than C1, then
we say that C1 is equivalent to C2.
Definition 1.13 If C1 is not weaker than C2 and C2 is not weaker than C1, then we say that C1 and C2 are not comparable.
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Example 8
Let X = {a, b, c} and τ on X be τ = {, X,{a},{c},{a, c}}. Let C1 =
{τ11, τ12, τ13} where τ11 = {, X, {c}, {a, c}}, τ12 = {, X, {c}}, and τ13 = {, X}. Then C1 is a chain of strong reductions of τ.
Let C2 = {τ21, τ22, τ23} where τ21 = {, X,{a},{a, c}}, τ22 = {, X,{a ,c}}, and τ23 = {, X}. Then C2 is another chain of strong
reductions of τ.
We see that C1 and C2 are not comparable because the topology τ12 in C1 is not comparable to any topology in C2; and τ21 in C2 is not
comparable to any topology in C1.
Example 9
Let C1 remain as in the example above and let C3 = {τ31, τ32, τ33} where τ31 = {, X, {c}, {a, c}}, τ32 = {, X,{a, c}}, and τ33 = {,
X}. Then C3 is another chain of strong reductions of τ and we see that C1 is weaker (but not strictly) than C3, since every topology
in C1 is weaker than τ31. And if we also observe that every topology in C3 is weaker than τ11, then we know that C1 and C3 are
equivalent.
Example 10
Let (R, u) denote the set of real numbers with its usual topology. Let Z denote the set of integers. For each z Z, let Xz be the u-
open interval Xz = (−∞, z). Then clearly
{G u : G Xz} = {G u : G (−∞,z)}
is a topology on Xz. Let τz = Xz-topology on R; in that τz = {G u : G Xz}{R} = {G u : G (−∞,z)}{R}.
Then clearly if z1 < z2, we have and τz1 is weaker than τz2. Hence the family
is a chain of weak reductions of the usual topology on R, in that
··· < τz−2 < τz−1 < τz0 < τz1 < τz2 < ··· < u,
where u is the usual topology on R.
For each n N (= the set of natural numbers), let Xn = (−n, n) and let τn = {G u : G Xn}{R} be an Xn-topology of R, obtained
from the usual topology on R. For instance, X1 = (−1,1) and τ1 = {G u : G X1}{R} is an X1-topology on R strictly weaker than
the usual topology on R. Also X2 = (−2, 2) and τ2 = {G u : G X2}{R} is an X2-topology of R obtained from the usual topology
on R. And so on. Then
is a chain of weak reductions of u. Since, for each n IN, the set (−n, n) is a proper subset of (−∞,n), and we see that the chain
is strictly weaker than the chain
.
What happens on a weak topology in terms of reducibility? We will now show that if τ is a weak topology on a set X, and one of
the range spaces of (X, τ) is reducible in the strong sense, then there exists a chain of weak topologies, each weaker than τ, on X
(generated by the fixed family of functions), which are a chain of reductions of τ (not necessarily in the strong sense) if the function
associated with the strongly reducible range space has requisite properties. We prove this next in a theorem.
The following lemma will be useful in the theorem that follows after.
Lemma 1.1 If τ is a topology on X and τ1 = τ {G} is a topology on X
(where G τ), then τ1 is only one set, G, strictly finer than τ.
Proof:
τ1 is a strong extension of τ and is, hence, only one set strictly finer than τ.
■
Note
What Lemma 1.1 says is that the introduction of just one set G into a topology τ to produce another topology τ1 does not make τ1 to
have more than one open set (either from finite intersections or arbitrary unions) than τ—and that the extra open set is precisely G.
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Theorem 1.2 Let (X, τ) be a weak topological space generated by the family {(Xα,τα)} of topological spaces, together with the family
{fα} of functions. There exists a chain of weak topologies, each weaker than τ, on X (generated by this fixed family of functions),
which are a chain of reductions of τ if (a) one of the range spaces, say τα, has a chain of strong reductions, (b) fα is one-to-one, and
(c) fα maps into all the elements of each topology in the chain of strong reductions of τα.
Proof:
Let (Xα,τα) be the range space meeting the hypotheses, for some α ∆, and let
be a chain of strong reductions of τα. Let and be any two topologies in such that, say is strictly weaker than by one
set. That is, is a strong reduction of . Let
and
= {
(G2i) : G2i }.
Then clearly
as is closed under finite intersections. That is, τ1 is closed under finite intersections. Also
,
implying that τ1 is closed under arbitrary unions. It is easy to see that , X τ1 as , Xα . Hence τ1 is a topology on X,
corresponding to . Similarly τ2 is a topology on X corresponding to . It is easy to see that both τ1 and τ2 are weaker than τ.
It is obvious that τ1 is weaker than τ2 and (by Lemma 1.1) that τ1 is only one set less than τ2. That is, τ1 is a strong reduction of τ2.
As τr1 and τr2 in are arbitrary it follows that there corresponds to a chain C of topologies on X of pair-wise comparable
topologies which can be arranged in such a way that each one is strictly weaker than the next by only one set. If we let the elements
of C to represent the (hypothetical) range space (Xα,τα)—one after the other—in the collection of sub-base for weak topologies on
X while leaving the other range spaces unchanged, the required chain of weaker weak topologies on X will emerge.
■
II. Summary & Conclusions
1. The concepts of strong, normal and weak reduction of topologies are introduced.
2. We proved that the discrete topology of a set X cannot be reduced in the strong sense of the cardinality of X is greater than
2.
3. We proved that the usual topology of the set of real numbers cannot be reduced in the strong sense.
4. The concepts of base reduction and strong base reduction of topologies are introduced.
5. The concepts of strong and weak extensions of topologies are introduced.
6. Strong base extension, weak base extension and base extension of topologies are introduced.
7. We established the conditions which guarantee that a topology is extensible, weakly extensible, base or weakly base
extensible.
8. The idea of a chain of reductions for a topology is introduced, as well as the idea of comparable and equivalent chains of
reductions.
9. We obtained the conditions for a weak topology to have a chain of reductions.
10. Ample examples are given at appropriate places to illustrate the ideas discussed.
Theorem 1.2 indicates that a fixed family of functions can generate a family of pairwise comparable weak topologies. Further
research may now embark on finding more considerations for this result. This is part of the developments in our published works
titled Comparison Theorems for Weak Topologies (see references below).
Note
So far, all the chains of strong reduction of topologies given in this paper are countable. The question then arises as to whether there
can be an uncountable chain of strong reductions of some topology. For example, can an uncountable chain of strong reductions be
obtained for the usual topology of R? Further, if a range topology for a weak topology has an uncountable chain of strong reductions,
what is the implication of this on the weak topology? That is, does the weak topology in this case inherit this property? Can we
characterize the weak topologies for which there exist families of other weak topologies which are chains of strong reductions of
the given weak topologies? Answers to these questions are as yet unknown and open a window for further research in this area.
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