Comparison Theorems for Weak Topologies (1)

Comparison Theorems for Weak Topologies (1)

¹Chika S. Moore., ¹Alexander O. Ilo., ²Ifeanyi Omezi

¹Department of Mathematics, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State

²Department of Petroleum Engineering, Nnamdi Azikiwe University, P.M.B. 5025, Awka, Anambra State

DOI: https://doi.org/10.51584/IJRIAS.2024.908060

Received: 10 July 2024; Revised: 25 July 2024; Accepted: 29 August 2024; Published: 18 September 2024

ABSTRACT

Weak topology on a nonempty set X is defined as the smallest or weakest topology on X with respect to which a given (fixed) family of functions on X is continuous.

Let τ_w be a weak topology generated on a nonempty set X by a family {f_α,:α ∈ ∆} of functions, together with a corresponding family {(X_α, τ_α):α ∈ ∆} of topological spaces. If for some α₀ ∈ ∆, τ_α0 on X_α0 is not the indiscrete topology and f_α0 meets certain requirements, then there exists another topology τ_w1 on X such that τ_w1 is strictly weaker than τ_w and f_α is τ_w1-continuous, for all α ∈ ∆. Here in Part 1 of our Comparison Theorems for Weak Topologies it is observed that: (a) the new topology τ_w1 on X deserves to be called a weak topology (with respect to the fixed family of functions) in its own right. Hence, we call τ_w1 a strictly weaker weak topology on X, than τ_w; (b) the usual weak, weak star, and product topologies have chains of pairwise strictly comparable (respectively) weak, weak star, and product topologies around them.

All the necessary and sufficient conditions for the existence of τ_w1 in relation to τ_w are established. Ample examples are given to illustrate (at appropriate places) the various issues discussed.

Key Words: Topology, Weak Topology, Weak Topological System, Strictly Weaker Weak Topologies, Pairwise Strictly Comparable Weak Topologies

Mathematics Subjects Classification (MSC) 2020: 54A05, 54A10

INTRODUCTION

The word Topology in Mathematics is comparable to the meaning of the word Topography in either Geography or Geology. While topography means ‘the observable nature of a landscape’ in both geography and geology, topology means the nature of the mathematical landscape upon which mathematical activities and operations (such as analysis of functions) can be performed. For example, if the topography of a land area is good, we can consider building a residential house on that piece of land; but if the land piece is an area of active volcano or a waterlogged swampy place, no one may think of going there to build a house. In mathematics, the effectiveness or efficacy of a function that we may define on a set depends on the topology of both the domain set and the range set. One mathematical set can have several topologies (plural of topology). For instance, a three-element set has 29 topologies, a 4-element set has exactly 355 topologies, a 7-element set has 9,535,241 topologies, and so on. Then mathematically speaking, what is a topology? Our first definition answers this question.

The topography of a geographical space determines the socioeconomic activity that can be carried out on that space. Similarly, the topology of a mathematical space (a set) determines the kind of mathematical activity that can be carried out on the space. Turn the focus around and ask the question: Can the socioeconomic activity (or the mathematical activity) on a geographical area (resp. mathematical space) determine the topography (the topology) of the geographical area (of the mathematical set)? The answer is ‘Yes’. Our aim in this study is not only to use constructive approach and practical examples to show that several weak topologies can be determined by a fixed family of functions on their common domain, but also to show how these topologies compare with one another.

The title of this research paper is: Comparison Theorems for Weak Topologies (1). The bracketed (1) here means that this research culminates into more than one article and that this manuscript is the number 1 in the entire three-part series of research work.

Definition 1.1 Let X be a nonempty set, and let 𝜏 be a collection of subsets of X in such a way that𝜏 contains the empty set, the entire set X , the intersection of any finite number of sets in 𝜏 and contains the union of any number of sets in 𝜏.  Then 𝜏 is called a topology on X.

Definition 1.2 If 𝜏₁ and 𝜏₂  are two topologies on X such that (say) every set in 𝜏₁ is in 𝜏₂ , then we say that the topology 𝜏₁ is weaker than the topology 𝜏₂. If 𝜏₁  is weaker than 𝜏₂ and there exists a set in 𝜏₂ which is not in 𝜏₁ we say that 𝜏₁ is strictly weaker than 𝜏₂.

Definition 1.3 If 𝜏_w is the weak topology on X generated by the family {(X_α, τ_α)}_α∈∆ of topological spaces, together with the family {f_α}_α∈∆ of functions, we shall call the triple [(X,𝜏_w), {(X_α,τ_α)}_α∈∆, {f_α}_α∈∆] a weak topological system.

Definition 1.4 A product topological system is a triple [(X, 𝜏_p), {(X_α, τ_α)}, {p_α}] _α∈∆ of a topological product space (X, 𝜏_p), a family of topological spaces {(X_α,τ_α)} which, together with the family {p_α} of projection maps, induce the product topology 𝜏_p on X^¯.

We observe that every product topological system is a weak topological system but the converse is not true.

Definition 1.5 Let [(X, 𝜏_w), {(X_α, τ_α)} _α∈∆, {f_α}_α∈∆] be a weak topological system. The weak topology 𝜏_w is called an indiscrete weak topology (or a minimal weak topology)^[1] if the family of functions in this system cannot generate a strictly weaker weak topology than 𝜏_w, on X.

Definition 1.6 Let [(X, 𝜏_w), {(X_α, τ_α)} _α∈∆, {f_α}_α∈∆] be a weak topological system. The weak topology 𝜏_w is called a discrete weak topology (or a maximal weak topology)^[2] if the family of functions in this system cannot generate a strictly stronger weak topology than 𝜏_w, on X.

MAIN RESULTS—SOME PRELIMINARY DEVELOPMENTS

Lemma 2.1 Let Ψ and Φ be two nonempty subsets of the power set 2^X of a nonempty set X such that (say) Ψ is a proper subfamily of Φ. If f is a 1-1 function mapping into all the elements of Φ, and there exists an element of the domain of f mapped into an element of Φ not in Ψ, then S₁ = {f⁻¹(G) : G ∈ Ψ} is a proper subfamily of S₂ = {f⁻¹(G) : G ∈ Φ}.

Proof:

Ψ is a proper subfamily of Φ. So, there exists G₀ ∈ Φ ∋ G₀ ∉ Ψ, and G ∈ Φ∀G ∈ Ψ. Therefore, from this hypothesis S₁ = {f⁻¹(G) : G ∈ Ψ} is a subfamily of S₂ = {f⁻¹(G) : G ∈ Φ} and—since G₀ ∉ Ψ and f is 1-1—in particular the set f⁻¹(G₀) ∉ S₁ (for otherwise we will have a contradiction). This means that S₁ is a proper subfamily of S₂. ⊙

Remark

If f is not 1-1, S₁ may equal S₂ even though Ψ is a proper subfamily of Φ. See examples 1 and 2 below. And if f is 1-1 and there is no element of the domain of f mapped into an element of Φ not in Ψ, then again S₁ may equal S₂; this is illustrated in example 3.

Example 1:

Let X = {a, b, c}, Ψ = {∅, X, {a}} and Φ = {∅, X, {a}, {b}, {a, b}}, and let E be any nonempty set with cardinality greater than 1. Let f: E → X be a map such that f(E) = {a}. Then S₁ = {f⁻¹(G): G ∈ Ψ} = {∅, E} and S₂ = {f⁻¹(G): G ∈ Φ} = {∅, E}. That is, S₁ = S₂.

Example 2:

Let E = {1,2,3.4,5,6}, X = {a, b, c, d} and Let g: E → X be a map such that g(1) = a = g(4); g(3) = b = g(5). Let Ψ = {∅, X, {a}, {b}, {a, b}} and Φ = {∅, X, {a}, {b}, {a, b}, {c}, {a, b, c}}. Then Ψ is a proper subfamily of Φ but S₁ = {g⁻¹(G): G ∈ Ψ} = {∅, {1,3,4,5}, {1,4}, {3,5}} and S₂ = {g⁻¹(G): G ∈ Φ} = {∅, {1,3,4,5}, {1,4}, {3,5}}. So S₁ = S₂. We see that g⁻¹(X) = g⁻¹({a, b}) = g⁻¹({a, b, c}) = {1,3,4,5}.

Example 3:

Let X = {a, b, c}, Ψ = {∅, X, {a}} and Φ = {∅, X, {a}, {b}, {a, b}}. Let f: E → X be a map such that f(E) = {a}, where E is a singleton. Then f is 1-1 but S₁ = {f⁻¹(G): G ∈ Ψ} = {∅, E} and S₂ = {f⁻¹(G): G ∈ Φ} = {∅, E}. That is, S₁ = S₂.

Example 4:

Let E, X, Ψ and Φ all be as defined in example 2 and let g: E → X be a map defined by g(1) = a, g(3) = b, g(4) = c, g(5) = d. We now have g⁻¹(∅) = ∅, g⁻¹(X) = {1,3,4,5}, g⁻¹({a}) = {1}, g⁻¹({b}) = {3} and g⁻¹({a, b}) = {1,3}. Therefore, S₁ = {g⁻¹(G): G ∈ Ψ} = {∅, {1,3,4,5}, {1}, {3}, {1,3}}.

Now g⁻¹({c}) = 4 and g⁻¹({a, b, c}) = {1,3,4}. Hence S₂ = {g⁻¹(G): G ∈ Φ} = {∅, {1,3,4,5}, {1}, {3}, {1,3}, {4}, {1,3,4}}. We now see that S₁ is a proper subfamily of S₂.

Henceforth whenever we mention 1-1 function in a weak topological system we shall assume that it meets the conditions of lemma 2.1; except otherwise stated.

Proposition 2.1 Let [(X, 𝜏_w), {(X_α, τ_α)}_α∈∆, {f_α}_α∈∆] be a weak topological system. For some α_o ∈ ∆, arbitrary but fixed, let 𝜏_o be a topology on such that 𝜏_o is strictly weaker than . If (for this fixed α_o ∈ ∆) f_α0 is 1-1, then ∃τ_w1, a topology on X, such that (i) τ_w1 < 𝜏_w and (ii) f_α is continuous with respect to τ_w1, for all α ∈ ∆.

Proof:

Let

and let

Then by lemma 1, S₁ is a proper subfamily of S₂ since 𝜏_o  is strictly weaker than  and  is 1-1. We know that S₂ is a sub-base for 𝜏_w; and similarly, since 𝜏_o is a topology on , S₁ is a sub-base for another topology 𝜏_w1 on X.  As S₁ is a proper subfamily of S₂, there exists at least one set, say G, in S₂ such that G ∉ S₁. It follows those finite intersections of sets in S₂ (that is, base for 𝜏_w) contains at least one set G more than the finite intersections of the sets in S₁ (which is a base for 𝜏_w1). Hence the topology 𝜏_w1 is weaker than 𝜏_w by at least one set G. That is, 𝜏_w1 is strictly weaker than 𝜏_w. We also observe that f_α is 𝜏_w1 -continuous, for each α ∈ ∆.

⊙

Observations:

1. The proposition above and the lemma 2.1 that facilitated its proof relied heavily on the existence of just one 1-1 function in a weak topological system, not on the existence of 𝜏_o; since every non-indiscrete topology has a strictly weaker topology.

2. Two weak topologies almost always the only ones of interest (so-called the weak and the weak star topologies) to many authors are about linear maps on linear spaces. The questions now vis-a-vis the proposition 2.1 here are Is every linear map a 1-1 function? The answer is ’No’. Projection maps are linear but not 1-1.

Does there exist linear maps which are 1-1? Answer: ’Yes’. The identity maps are linear and 1-1.

Is every 1-1 map linear? Answer: ’No’. The function f(x) = x³ is 1-1 but not linear.

3. Since there exist linear maps which are 1-1 and since the usual weak and weak star topologies are general statements about linear maps, proposition 2.1 implies that these topologies have strictly weaker weak or weak star topologies.

4. Among the results represented by the exposition of this paper is the fact that the usual weak and weak star topologies, among others, have chains of pairwise strictly comparable weaker weak topologies.

Corollary 2.1 The usual weak and weak star topologies have chains of pairwise strictly comparable weaker weak or weak star topologies.

Proof:

Since these topologies are weak topologies generated, on sets (linear spaces precisely), by all the linear maps on such sets, since some linear maps (namely, the identity maps) are 1-1 functions, Proposition 2.1 ensures this result.

⊙

It may appear by now that it is only when a function f is 1-1 that S₁ would be a proper subfamily of S₂ given that Ψ is a proper subfamily of Φ. This is not so. In fact, f being 1-1 is only a sufficient condition for S₁ to be a proper subfamily of S₂ (given that Ψ is a proper subfamily of Φ) but it is not a necessary condition. The following example illustrates this.

Example 5:

Let E, X, Ψ and Φ all be as given in examples 2 and 4 above. Let h: E → X be a map defined by h(1) = a, h(2) = c, h(3) = b, h(4) = a and h(5) = b. Then we see that

S₁ = {h⁻¹(G): G ∈ Ψ} = {h⁻¹(∅), h⁻¹(X), h⁻¹({a}), h⁻¹({b}), h⁻¹({a, b})} =

{∅, {1,2,3,4,5}, {1,4}, {3,5}, {1,3,4,5}}. And that

S₂ = {h⁻¹(G): G ∈ Φ}

= {h⁻¹(∅), h⁻¹(X), h⁻¹({a}), h⁻¹({b}), h⁻¹({a, b}), h⁻¹({c}), h⁻¹({a, b, c})} = {∅, {1,2,3,4,5}, {1,4}, {3,5}, {1,3,4,5}, {2}}.

We observe that card(S₁) = 5 and card(S₂) = 6; that S₁ ⊂ S₂ and that S₁ ≠ S₂. A more general form of lemma 2.1 can therefore be stated as follows.

Lemma 2.2 Let Ψ and Φ be two nonempty subsets of the power set 2^X of a nonempty set X such that Ψ is a proper subfamily of Φ. If f is a function mapping into each element of Φ, and there exists G₀ ∈ Φ − Ψ such that f⁻¹(G₀) ≠ f⁻¹(G), ∀G ∈ Ψ, then S₁ = {f⁻¹(G): G ∈ Ψ} is a proper subfamily of S₂ = {f⁻¹(G): G ∈ Φ}.

Proof:

Since ∃G₀ ∈ Φ, ∋ f⁻¹(G₀) ≠ f⁻¹(G), ∀G ∈ Ψ and since Ψ ⊂ Φ it follows that the collection S₁ = {f⁻¹(G): G ∈ Ψ} is a proper subfamily of S₂ = {f⁻¹(G): G ∈ Φ}.

⊙

We can now also obtain a more general form of proposition 2.1.

Proposition 2.2 Let [(X, 𝜏_w), {(X_α, τ_α)}_α∈∆, {f_α}_α∈∆] be a weak topological system. For some α₀ ∈ ∆, arbitrary but fixed, let τ₀ be a topology on X_α0 such that τ₀ is strictly weaker than τ_α0. If ∃G₀ ∈ τ_α0 such that

 ,

then ∃τ_w1, a topology on X, such that (i) τ_w1 < 𝜏_w and (ii)

f_α is continuous with respect to τ_w1, for all α ∈ ∆.

Proof:

Since ∃G₀ ∈ τ_α0 such that , it follows that G₀ ∈ τ_α0 − τ₀ and (by lemma 2) in particular

is a proper subfamily of

.

Clearly elements of S₂ are among the sub-basic sets of 𝜏_w   and, since τ₀ is a topology, S₁ is also a subset of a sub-base for another topology τ_w1 on X, strictly weaker than 𝜏_w. Since [(X, τ_w1), {(X_α, τ_α)}_α∈∆, {f_α}_α∈∆] is a weak topological system, f_α is τ_w1-continuous, ∀α ∈ ∆.

⊙

Remark:

Proposition 2.2 implies that even a product topology can have a strictly weaker product topology.

EXAMPLE 6:

Let X₁ = {a, b} = X₂ be two sets and let  =  = {(a, a), (a, b), (b, a), (b, b)}.

Let the projection maps on be defined in the usual way  by  p_i{(x, y)} = x, if i = 1 and p_i{(x, y)} = y, if i = 2. Let both factor spaces of  be endowed with the topology τ = {∅, {a}, {b}, {a, b}}. Then the product topology 𝜏_p on is 𝜏_p = 2^x, the power set of ; which is a family of 16 subsets of .

If we now let a factor space of , say X₁, be endowed with a topology τ₀ strictly weaker than τ such that ∃G₀ ∈ τ and such that  we shall get a strictly weaker product topology , on , than 𝜏_p. To see this, let τ₀ on X₁ be τ₀ = {∅, X₁, {a}}. Then (with the topology of X₂ still being τ) the product topology now on  would be   a family of only 9 subsets of .

It can also be verified easily that both projection maps p₁ and p₂ are continuous with respect to  if τ₀ and τ are endowed on X₁ and X₂ respectively.

Note:

Example 6 actually represents a general phenomenon in product topological systems; namely that if [(X, 𝜏_p), {(X_α,τ_α)}, {p_α}]_α∈∆ is a product topological system, and there exists  ∈ ∆ such that  has a strictly weaker topology , on  then there exists a strictly weaker product topology τ_p1 than 𝜏_p on X¯ with respect to which all the projection maps are continuous. We shall give a formal proof of this later, but for now, let’s have another lemma.

Lemma 2.3 Let p_α:  → X_α be a projection map of a Cartesian product set onto a factor space. If x_α1 and x_α2 are two different elements of X_α, then ) ≠ ).

Proof:

Since projection maps count coordinates and return them to respective (or corresponding) factor spaces, we have

.

Also

.

As tuples (or vectors) are equal if and only if their corresponding components are equal, and since x_α1 ≠ x_α2, we must have ; that is,  ) and ) have no element in common. As both ) and ) are nonempty, it follows that ) ≠).

⊙

Corollary 2.2 Let  be a projection mapping. If A and B are two nonempty subsets of X_α such that (say) A is a proper subset of B, then  and  ; that is, is a proper subset of  .

Proof:

Since A ⊂ B and A ≠ B, ∃b₀ ∈ B ∋ b₀  A. This implies that b₀ ≠ a, ∀a ∈ A. This implies (by lemma 2.3) that , ∀a ∈ A. This implies that .

But , because A ⊂ B. And we also know that  as b₀ ∈ B. Hence is a proper subset of  .

⊙

Corollary 2.3 Let  be a projection mapping and let Ψ and Φ be two nonempty subsets of the power set of  X_α. If Ψ is a proper subfamily of Φ, then  is a proper subfamily of

.

Proof:

Clearly  is a subfamily of , from hypothesis. We only show that S₁ ≠ S₂. Let G₀ ∈ Φ − Ψ. Since each set is the union of its own elements, we have

This implies that . This implies that and since , it follows that S₁ ≠ S₂. That is, S₁ is a proper subfamily of S₂.

⊙

Proposition 2.3 Let [(X, 𝜏_p), {(X_α, τ_α)}, {p_α}]_α∈∆ be a product topological system. If (for some α₀ ∈ ∆) τ_α0 has a strictly weaker topology τ₀, on X_α0, then the product topology 𝜏_p on X^¯ has a strictly weaker product topology, .

Proof:

From hypothesis  is a proper subfamily of . By corollary 2.3,  is a proper subfamily of . Since  is a topology on , it follows that is part of a sub-base for a product topology  on  (with the topologies of the other factor spaces unchanged). Since S₂ is part of a sub-base for 𝜏_p and since S₁ is a proper subfamily of S₂,   is strictly weaker than 𝜏_p.

⊙

Remark:

1. It is now clearer that the condition of 1-1-ness in proposition 2.1 is only a sufficient, but not necessary, requirement for a strictly weaker weak topology to be obtained, given that the topology of a range space has a strictly weaker topology.
2. The reasoning in propositions 2.1 and 2.2 implies that if τ₁ is strictly weaker than τ₀, τ₂ strictly weaker than τ₁, and so on, then there exist correspondingly weak topologies τ_w2, τ_w3, etc., on X, such that 𝜏_w > τw1 > τw2 > τw3 > ··.
3. If we have a weak topological system [(X, 𝜏_w), {(X_α, τ_α)}_α∈∆, {f_α}_α∈∆], one question is whether we can always find another topology τ_w1 on X such that 𝜏_w > τ_w1 and such that each function in the family is continuous? That is, does τ_w1 always exist for every weak topology 𝜏_w? Another question (if it be found that τ_w1 does not exist for all weak topologies 𝜏_w) is whether we can characterize such weak topologies 𝜏_w for which we can find such τ_w1. And yet another question is: What (if any) topological property can 𝜏_w transmit to, or induce on τ_w1? This last question can be seen as property inheritance question—and it is as important here as it is in human society. These questions and more are what we shall be looking at in the next section.

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FOOTNOTES

[1] As we shall see later, an indiscrete weak topology on X may not equal what may, now, be called the ordinary indiscrete topology {X, ∅} of X.

[2] As we shall see later, a discrete weak topology on X may be strictly weaker than what may be called the ordinary discrete topology {X, 2^X} of X.