AN INTERESTING APPLICATION OF CLASSICAL TOPOLOGY IN REAL ANALYSIS

Dušan Repovš

Institute of Mathematics Physics and Mechanics

Jadranska 19, P. O. Box 2964 Ljubljana, Slovenia 1001

e-mail: dusan.repovs@fmf.uni-lj.si 1. Introduction Recall the ε-δ definition of continuity of a map f : (X, d) → (Y, ρ) between metric spaces:

(∀x ∈ X)(∀ε > 0)(∃δ > 0)(∀x′ ∈ X)

d(x, x′) < δ ⟹ ρ(f(x), f(x′)) < ε i.e. there exists a (not necessarily continuous) function δ : X × C(X, Y) × IR_{+}^{∗} → IR_{+}^{∗} such that d(x, x′) < δ(x, f, ε) ⟹ ρ(f(x), f(x′)) < ε, for all values of variables

x, x′ ∈ X, f ∈ C(X, Y) and ε ∈ IR_{+}^{∗}. Here C(X, Y) denotes the space of all continuous maps from X to Y, endowed with the following metric of uniform convergence: ρ_{τ}(f, g) = sup{min{1, ρ(f(x), g(x))}|x ∈ X}.

Let each such function δ be called a modulus of continuity. The following was the first result on existence of continuous moduli of continuity: Theorem 1 (Repovš–Semenov) Let f : (X, d) → (Y, ρ) be a continuous map between metric spaces and X a locally compact space. Then there exists a continuous function δ̂: X × C(X, Y) × IR_{+}^{∗} → IR_{+}^{∗} such that for every (x, f, ε) ∈ X × C(X, Y) × IR_{+}^{∗} and every x′ ∈ X the following holds:

d(x, x′) < δ̂(x, f, ε) ⟹ ρ(f(x), f(x′)) < ε.

Consider two elementary applications of this theorem. First, recall two classical results from analysis: Let f : X → IR be a continuous function on a compact metric space X.

Then the Weierstrass theorem asserts that f is bounded (above and below) and the Cantor theorem asserts that f is uniformly continuous on X. Corollary 1 The Cantor theorem is a corollary of the Weierstrass theorem. Proof. Suppose that f : (X, d) → IR is continuous and X compact. Apply Theorem 1 to get a continuous modulus of continuity δ̂ : X × C(X, Y) × IR_{+}^{∗} → IR_{+}^{∗}. For a fixed f ∈ C(X, Y) and ε ∈ IR_{+}^{∗} we can now choose δ > 0 as follows

δ = min{δ̂(x, f, ε)|x ∈ X}, since by the Weierstrass theorem such minimum exists. Hence f is uniformly continuous.

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Corollary 2 Let G ⊂ C(X, Y) be a subset such that its closure Cl(G) is compact. Then G is an equicontinous set of maps. Proof. Recall that G is said to be equicontinuous if for each ε ∈ IR_{+}^{∗} there exists δ ∈ IR_{+}^{∗} such that for every f ∈ G and x, x′ ∈ X

d(x, x′) < δ ⟹ ρ(f(x), f(x′)) < ε.

So under the hypothesis, it suffices to pick, for a fixed ε ∈ IR_{+}^{∗}, our δ > 0 as follows

δ = min{δ̂(x, f, ε)|x ∈ X, f ∈ Cl(G)}.

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2. Alternative proofs of Theorem 1

A map u : Z → IR is called upper (resp. lower) semicontinuous at z ∈ Z if for every ε > 0 there exists a neighbourhood U ⊂ Z of z such that u(z′) < u(z) + ε (resp. u(z′) > u(z) − ε), for each z′ ∈ U.

Example. The maps u, v : IR → IR given by

u(z) = \{ \begin{matrix}4; & if z = 2 \\ 3; & if z ≠ 2 \end{matrix} v(z) = \{ \begin{matrix}1; & if z = 2 \\ 2; & if z ≠ 2 \end{matrix}

are upper (resp.) lower semicontinuous.

[figure: ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; z; u; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; z; v]

Theorem 2 (Dowker) Let u, v : Z → IR be any functions, defined on a paracompact space Z, such that u(z) > v(z), for every z ∈ Z. Assume also that v is upper semicontinuous and u is lower semicontinuous. Then there exists a continuous w : Z → IR such that u(z) > w(z) > v(z), for every z ∈ Z.

Proof of Theorem 1(using Dowker’s theorem). Denote by B_{δ}(x) the open ball of radius δ with center x ∈ X. Introduce a function Δ : X × C(X, Y) × IR_{+}^{∗} → IR as follows

Δ(x, f, ε) = sup{δ > 0|Cl(B_{δ}(x)) is a compact

set and d(x, x′) < δ ⟹ ρ(f(x), f(x′)) < ε}, for each x′ ∈ X.

By local compactness of X, Δ is well-defined and lower semicontinuous. Apply Dowker’s Theorem to

Δ : X × C(X, Y) × IR_{+}^{∗} → IR

and the zero-function

X × C(X, Y) × IR_{+}^{∗} → {0}

to obtain a continuous function

δ̂: X × C(X, Y) × IR_{+}^{∗} → IR_{+}^{∗}

such that Δ(x, f, ε) > δ̂(x, f, ε) > 0.

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A multivalued map F : X → Y is said to be lower semicontinuous if for every open subset G ⊂ Y the set {x ∈ X|F (x) ∩ G ≠ ∅} is open in X. A singlevalued map f : X → Y is called a selection of a multivalued map F : X → Y if f(x) ∈ F (x), for each x ∈ X. A convex subset of a Banach space is said to be of D-type if it contains all interior (in the convex sense) points of its closure. A point of a closed convex subset of a Banach space is said to be interior (in the convex sense) if it is not contained in any supporting hyperspace. It is easy to see that all finite-dimensional convex sets are examples of convex D-type sets. In this case the set of all interior (in the convex sense) points coincides with the set of all interior (in the usual sense) points.

Theorem 3 (Michael) Let Z be a perfectly normal space and B a separable Banach space. Then each lower semicontinuous map F : Z → B with convex D-type images has a continuous selection f : Z → B. Proof of Theorem 1(using Michael’s theorem). Let F : X ×C(X, Y)×IR_{+}^{∗} → IR_{+}^{∗} be a multivalued map, given by F (x, f, ε) = (0, Δ(x, f, ε)], where Δ was defined earlier. Then F is lower semicontinuous since Δ is lower semicontinuous. Furthermore, each interval is a convex D-type set and the space X ×C(X, Y)×IR_{+}^{∗} is perfectly normal since it is metrizable. Apply Michael’s Theorem to get a continuous selection δ̂ : X × C(X, Y) × IR_{+}^{∗} → IR_{+}^{∗} such that δ̂(x, f, ε) ∈ (0, Δ(x, f, ε)].

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3. Continuity-like predicates

Proof of Theorem 1 (without the hypothesis of local compactness of X, due to Artico and Marconi). Introduce the sets: Ctrue = {(x, x′, f, ε) ∈ X×X×C(X, Y)×IR_{+}^{∗}|

ρ(f(x), f(x′)) < ε}

Cfalse = {(x, x′, f, ε) ∈ X×X×C(X, Y)×IR_{+}^{∗}|

ρ(f(x), f(x′)) ≥ ε}.

The function (x, x′, f, ε) ↦ ρ(f(x), f(x′))−ε is continuous, hence the set Ctrue is open and thus Cfalse is closed. Obviously (x, x, f, ε) ∈ Ctrue. Therefore, the function δ̂: X ×C(X, Y)×IR_{+}^{∗} → IR_{+} defined by: δ̂(x, f, ε) = dist((x, x, f, ε), Cfalse) is strictly positive. It remains to verify that δ̂ is continuous.

This proof gives an idea how to generalize Theorem 1. Let F(X, Y) be the space of all (not necessarily continuous) maps, equipped with the norm of uniform convergence. Let

P : X × X × F(X, Y) × IR_{+}^{∗} → {true, false}

be a logical predicate and let P true = P^{−1}(true) and P false = P^{−1}(false). Definition 1 If for each x ∈ X and ε ∈ IR_{+}^{∗} there exists a neighbourhood U ⊂ X of x such that (x, x′, f, ε) ∈ P true, for all x′ ∈ U then f is called P -continuous. If there exists at least one P -continuous map, then P is called a continuity-like predicate. Let F_{P} (X, Y) ⊂ F(X, Y) be the subspace of all P -continuous maps and P true_{0}=P true∩(X×X×F_{P} ×IR_{∗}^{+}).

Example. Define the predicate C by :

C(x, x′, f, ε) = (ρ(f(x), f(x′)) < ε),

where f : X → Y is a map from a topological space X into a metric space (Y, ρ). Then Ccontinuous maps are precisely the continuous maps, hence C is the predicate of the usual continuity.

Example. Define the predicate USC by:

USC(x, x′, f, ε) = (f(x′) < f(x) + ε),

where f : X → IR is a function defined on a topological space X. Then USC-maps coincides with the upper semicontinuous maps, hence USC is the predicate of upper semicontinuity.

Example. Analogously, define the predicate LSC by:

LSC(x, x′, f, ε) = (f(x′) > f(x) − ε),

where f : X → IR is a function defined on a topological space X. Then LSC is the predicate of lower semicontinuity.

Example. Let α : X → [0, ∞] be any function. Define the predicate C^{α} by:

C^{α}(x, x′, f, ε) = (ρ(f(x), f(x′)) < α(x) + ε),

where f : X → Y is a map from a topological space X into a metric space (Y, ρ). The predicate C^{α} induces the notion of αcontinuity, which is a generalization of the usual continuity, i.e. each continuous map is α-continuous for arbitrary α.

Moreover, if α_{1}(x) > α_{2}(x), for all x ∈ X then each α_{2}-continuous map is also α_{1}-continuous. For example, 0-continuity is the usual continuity and ∞-continuity is a void condition, i.e. each map is ∞-continuous.

Remark. All predicates listed above are continuity-like predicates.

Remark. Obviously, each point of the form (x, x, f, ε) ∈ X × X × F(X, Y) × IR_{+}^{∗} lies in the set P true. Therefore,

diag(X) × F(X, Y) × IR_{+}^{∗} ⊂ P true,

where diag(X) denotes the diagonal set

{(x, x)|x ∈ X} ⊂ X × X.

The following theorem is a direct generalization of Theorem 1: Theorem 4 (Malešič–Repovš) Let P be a continuity-like predicate. Then the following assertions are equivalent:

(i) There exists a continuous modulus of P - continuity, i.e. a function

δ̂: X × FP(X, Y) × IR_{+}^{∗} → IR_{+}^{∗}

such that for every

(x, x′, f, ε) ∈ X×X×FP(X, Y)×IR_{+}^{∗} → IR_{+}^{∗}

the following implication holds:

d(x, x′) < δ̂(x, f, ε) ⟹ (x, x′, f, ε) ∈ P true.

(ii) The set diag(X) × F P(X, Y) × IR_{+}^{∗} lies in the interior of the set P true_{0}.

The construction of the modulus

δ̂: X × F_{P} (X, Y) × IR_{+}^{∗} → IR_{+}^{∗}

in the proof of the preceding theorem is exactly as in the proof of Theorem 1:

δ̂(x, f, ε) = dist((x, x, f, ε), P false).

Corollary 3 There exists a modulus δ̂: X× C(X, Y) × IR_{+}^{∗} → IR_{+}^{∗} of C-continuity. Proof. We have shown (in the proof of Theorem 1) that the set Ctrue is open if f is continuous. Therefore the condition (ii) is fulfilled and there exists a continuous modulus of Ccontinuity.

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Recall that any function defined on a discrete metric space X is always continuous, hence every modulus on X is necessarily continuous. Corollary 4 Let X be a nondiscrete metric space. Then there are no continuous moduli of USC- (resp. LSC) -continuity on X. Proof. Let x_{0} ∈ X be a nonisolated point and define f : X → IR as follows:

f(x) = \{ \begin{matrix}1; & if x = x_{0} \\ 0; & if x ≠ x_{0} \end{matrix}

Clearly, f is upper semicontinuous, but discontinuous at x_{0}.

It is easy to see that the point (x_{0}, x_{0}, f, 0.5) cannot be an interior point of USCtrue. By Theorem 4 there are no continuous moduli of USC-continuity. Analogously one can prove that there are no continuous moduli of LSCcontinuity.

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Although a continuous modulus of USC-con-tinuity δ̂: X × C(X, Y) × IR_{+}^{∗} → IR_{+}^{∗} cannot exist as a function of three variables

x ∈ X, f ∈ F_{USC}(X, Y) and ε ∈ IR_{+}^{∗},

there exists a continuous modulus as a function of one variable ε ∈ IR_{+}^{∗} :

Corollary 5 For each upper semicontinuous function f : X → IR and each point x ∈ X there exists a continuous function δ̂: IR_{+}^{∗} → IR_{+}^{∗} such that

(∀x′)d(x′, x) < δ̂(ε) ⟹ f(x′) < f(x) + ε.

Proof. Choose x ∈ X and f ∈ F_{USC} and introduce a function δ̃: IR_{+}^{∗} → IR_{+}^{∗} by:

δ̃(ε) = inf{δ > 0|

(∀x′)d(x′, x) < δ ⟹ f(x′) < f(x) + ε}. Obviously, δ̃(ε) is a nondecreasing function of ε. There exists a piecewise constant nondecreasing function ˜ δ̃: IR_{+}^{∗} → IR_{+}^{∗} such that

˜δ̃(ε) ≤ δ̃(ε), for each ε ∈ IR_{+}^{∗},

e.g. set ˜ δ̃(\frac{1}{n}) = δ̃(\frac{1}{n}), for all n ∈ IN.

A continuous piecewise linear function δ̂ : IR_{+}^{∗} → IR_{+}^{∗} can be found such that δ̂(ε) ≤ ˜δ̃(ε), by the construction depicted below:

[figure: ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; ·; …; …; ε; δ̂(ε); ˜δ̃(ε)]

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Corollary 6 Let α : X → [0, ∞] be a nonlower semicontinuous function and (Y, ρ) a connected metric space of infinite diameter. Then there exist no continuous moduli of C^{α}-continuity.

Proof. Let α be nonlower semicontinuous at a point x_{0} ∈ X. Since Y is connected and has infinite diameter, it is possible to choose points y_{0}, y_{1} ∈ Y so that ρ(y_{0}, y_{1}) = α(x_{0}). Define a map f : X → Y by

f(x) = \{ \begin{matrix}y_{0}; & if x = x_{0} \\ y_{1}; & if x ≠ x_{0} \end{matrix}

It remains to prove that the point (x_{0}, x_{0}, f, ε) is not an interior point of the set C^{α,true} for certain values of ε.

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Up until to now, the modulus of continuity was a function of three arguments: the point x ∈ X, the function f ∈ F_{P} (X, Y) and the number ε ∈ IR_{+}^{∗}.

Theorem 4 can be generalized to obtain the modulus which depends on five arguments: the two additional arguments being the metrics d and ρ. Denote by M_{X} the space of all metrics on X, as a subspace of F(X×X, IR), and M_{Y} the space of all metrics on Y, as a subspace of F(Y × Y, IR), with the induced uniform convergence metric topology. Then a function

δ̂: X × F_{P} × IR_{+}^{∗} × M_{X} × M_{Y} → IR_{+}^{∗}

defined by the relation

d(x′, x) < δ̂(x, f, ε, d, ρ) ⟹
⟹ (x, x′, f, ε, d, ρ) ∈ P true,

for all

x, x′ ∈ X, ε ∈ IR_{+}^{∗}, f ∈ F_{P} , d ∈ M_{X}, ρ ∈ M_{Y} ,

is called a modulus of P -continuity.

Theorem 5 (Malešič–Repovš) There exists a continuous modulus of continuity, i.e. a continuous function

δ̂: X × IR_{+}^{∗} × C(X, Y) × M_{X} × M_{Y} → IR_{+}^{∗}

such that

d(x′, x) < δ̂(x, ε, f, d, ρ) ⟹
⟹ ρ(f(x′), f(x) < ε

for each x′, x ∈ X, f ∈ C(X, Y), d ∈ M_{X} and ρ ∈ M_{Y} .

Remark. Recently Weihrauch and Zheng found a computable modulus of continuity, in the sense of effective analysis.