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The point of continuity property, neighbourhood assignments and filter convergences

Ahmed Bouziad (2012)

Fundamenta Mathematicae

We show that for some large classes of topological spaces X and any metric space (Z,d), the point of continuity property of any function f: X → (Z,d) is equivalent to the following condition: (*) For every ε > 0, there is a neighbourhood assignment ( V x ) x ∈ X of X such that d(f(x),f(y)) < ε whenever ( x , y ) ∈ V y × V x . We also give various descriptions of the filters ℱ on the integers ℕ for which (*) is satisfied by the ℱ-limit of any sequence of continuous functions from a topological space into a metric space.

Weak continuity properties of topologized groups

J. Cao, R. Drozdowski, Zbigniew Piotrowski (2010)

Czechoslovak Mathematical Journal

We explore (weak) continuity properties of group operations. For this purpose, the Novak number and developability number are applied. It is shown that if ( G , · , τ ) is a regular right (left) semitopological group with dev ( G ) < Nov ( G ) such that all left (right) translations are feebly continuous, then ( G , · , τ ) is a topological group. This extends several results in literature.

Weakly continuous functions of Baire class 1.

T. S. S. R. K. Rao (2000)

Extracta Mathematicae

For a compact Hausdorff space K and a Banach space X, let WC(K,X) denote the space of X-valued functions defined on K, that are continuous when X has the weak topology. In this note by a simple Banach space theoretic argument, we show that given f belonging to WC(K,X) there exists a net {fa} contained in C(K,X) (space of norm continuous functions) such that fa --&gt; f pointwise w.r.t. the norm topology on X. Such a function f is said to be of Baire class 1.

θ -regular spaces.

Janković, Dragan S. (1985)

International Journal of Mathematics and Mathematical Sciences

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