iconti1.tex February 18, 2005

1 Equivalence of I-continuity and continuity at a point for real functions

Main result is Theorem 1 which answers the question posed on the last seminar. Throughout the paper we assume that I is an admissible ideal.

First we recall some basic notions from general topology (see e.g. [2]). Let X, Y be topological spaces. Then a map f: X → Y is continuous at a point x ∈ X if for every neighborhood V ⊆ Y of f(x) there exists a neighborhood U ⊆ X of x such that f[U] ⊆ V. A map f: X → Y is continuous if and only if it is continuous at each point of X.

We next recall some definitions from [4]. They are very straightforward generalizations of corresponding notions defined for real functions in [1].

Definition 1. Let I be an ideal in ℕ. A sequence (x_{n})_{n=1}^{∞} in a topological space X is said to be I-convergent to a point x ∈ X if

A(U) = {n : x_{n} ∉ U} ∈ I

holds for each open neighborhood U of x. We denote it by I-lim x_{n} = x.

If I = I_{f} is the Fréchet ideal (i.e. the ideal containing exactly all finite subsets of ℕ) then I-convergence coincides with the usual convergence.

Definition 2. Let I be an ideal in ℕ and X, Y be topological spaces and x_{0} ∈ X. A map f: X → Y is called I-continuous at x_{0} if for each sequence (x_{n})_{n=1}^{∞} in X

I-lim x_{n} = x_{0} ⇒ I-lim f(x_{n}) = f(x_{0})

holds.

Given a topological space X and a point a ∈ X, denote by X_{a} the space constructed by making each point, other then a, isolated with a retaining its original neighborhoods. (I.e. a subset U ⊆ X is open in X_{a} if and only if a ∉ U or there exists an open subset V of X such that a ∈ V ⊆ U.) The topological space X_{a} is called prime factor of X at the point a.

In what follows we examine the relationship between I-continuity of a function f: X → Y at a point a ∈ X and I-continuity of the function f: X_{a} → Y for topological spaces X, Y.

Lemma 1. Let X be a topological space, a ∈ X and (x_{n})_{n=1}^{∞} be a sequence of points of X. Then I-lim x_{n} = a in X if and only if I-lim x_{n} = a in X_{a}.

Proof. This is an easy consequence of the fact that neighborhoods of a are the same in both topological spaces X and X_{a}, so in the definition of I-limit we have precisely the same system of sets A(U) for both spaces.

By letting I_{f} = I we get:

Corollary 1. Let X be a topological space, a ∈ X and (x_{n})_{n=1}^{∞} be a sequence of points of X. Then lim x_{n} = a in X if and only if lim x_{n} = a in X_{a}.

Lemma 2. If x ∈ X is an isolated point of a topological space X (i.e. the set {x} is open in X) then any map f: X → Y is I-continuous at x.

Dôkaz. If U = {x} then A(U) = {n : x_{n} ≠ x} ∈ I. Therefore for each V ⊂ Y such that f(x) ∈ V we get A(V) = {n : f(x_{n}) ∉ V } ⊂ {n : f(x_{n}) ≠ f(x)} ⊂ {n : x_{n} ≠ x} and V belongs to I. Hence I-lim f(x_{n}) = f(x).

Proposition 1. Let X, Y be topological spaces and a ∈ X. Then f: X → Y is I-continuous at a if and only if f: X_{a} → Y is I-continuous.

Proof. The map f: X_{a} → Y is obviously I-continuous at each point different from a, since these points are isolated (Lemma 2). If follows from Lemma 1 that continuity of f: X → Y at a is equivalent to continuity of f: X_{a} → Y at a.

If X = ℝ then a map f: X → Y is continuous if and only if it is I_{f}- continuous. This implies

Corollary 2. Let Y be a topological spaces. A map f: ℝ → Y is continuous at a if and only if it f: ℝ_{a} → Y is continuous.

We recall notions of sequential and Fréchet spaces (see e.g. [2], a thorough survey of these types of spaces and some related properties and problems is

e.g. paper [3] available at http://xyz.lanl.gov/abs/math.GN/0412558). A

topological space X is called a sequential space if a set A ⊆ X is closed if and only if together with any sequence it contains all its limits. A topological space X is called Fréchet space if for every A ⊂ X and every x ∈ \overline{A} there exists a sequence (x_{n})_{n=1}^{∞} of points of A converging to X. Every first-countable (in particular every metric space) is a Fréchet space and every Fréchet space is a sequential space.

Proposition 2. The space ℝ_{a} for any a ∈ ℝ is Fréchet.

Dôkaz. We need to show that if x ∈ \overline{V} then there is a sequence of points of V converging to x. If x is an isolated point then x ∈ \overline{V} implies x ∈ V and we can use the constant sequence x_{n} = x. So it remains to solve the case x = a.

Let V ⊆ ℝ and a ∈ \overline{V} in ℝ_{a}. This means that each neighborhood of a contains a point from V. Neighborhoods of a in the topological space ℝ are the same as in the topological space ℝ_{a}, therefore a ∈ \overline{V} in ℝ. Since ℝ is a Fréchet space, there exists a sequence (x_{n})_{n=1}^{∞} of points of V which converges to a in the space ℝ. Using Corollary 1 we get that it converges to a in ℝ_{a} as well.

Proposition 3 ([4, Corollary 2.4]). Let X be a sequential space and let I be an admissible ideal. Let Y be a topological space and let f: X → Y be a map. Then the following statements are equivalent:

(1) f is continuous,

(2) f is I_{f}-continuous,

(3) f is I-continuous.

Theorem 1. Let a ∈ ℝ. A map f: ℝ → Y is continuous at a if and only if f: ℝ → Y is I-continuous at a.

Sketch of the proof. f: ℝ → Y is continuous at a ⇔ (Corollary 2) f: ℝ_{a} → Y is continuous ⇔ (Proposition 2 and Proposition 3) f: ℝ_{a} → Y is I-continuous ⇔ (Proposition 1) f: ℝ → Y is continuous at a.

Proposition 1 can be strengthened without any changes in the proof.

Proposition 4. If X is a Fréchet space then each prime factor X_{a} of X is a Fréchet space

Also the claim of Corollary 1 holds for any topological space X. This follows again from the fact that a has the same neighborhoods in both topological spaces X and X_{a}.

Proposition 5. Let X, Y be topological spaces and a ∈ X. A map f: X → Y is continuous at a if and only if f: X_{a} → Y is continuous.

Using these fact we arrive to conclusion that I-continuity, I_{f}-continuity and continuity at a point a ∈ X are equivalent for any Fréchet space X. It is natural to ask if the same holds for any sequential space.

References

[1] V. Baláž, J. Červeňanský, P. Kostyrko, and T. Šalát. I-convergence and Icontinuity of real functions. Acta Mathematica, Faculty of Natural Sciences, Constantine the Philosopher University Nitra, 5:43–50, 2002.

[2] R. Engelking. General Topology. PWN, Warsaw, 1977.

[3] A. Goreham. Sequential convergence in topological spaces. 2004. arXiv:math.GN/0204116.

[4] M. Sleziak. I-continuity in topological spaces. Acta Mathematica, Faculty of Natural Sciences, Constantine the Philosopher University Nitra, 6:115–122, 2003.