Interiority of real functions

Abstract

Conditions concerning openness of continuous real functions defined on topological spaces are studied. It is shown that for locally compact metrizable spaces interiority of the function at a point is equivalent to having no local extremum at this point if and only if the domain space is connected im kleinen at the point. Some related results are some obtained and open questions are posed. In this paper interrelations are studied between some conditions which concern openness of a real function (defined on a topological space), absence of any local extremum of this function at a given point, and local connectedness of the domain space. Additionally, although this investigation is related to the behaviour of real functions on metric or even on topological spaces in general, it has its roots in functional analysis. Namely, a theorem has been proved in [6] concerning sufficient conditions under which a Lipschitzian real function defined on a subset of a Banach space X has a Lipschitzian open extension over the whole space X . Subsequently, the third named author faced the problem of how to extend the above-mentioned result to (either all or some) topological or metric spaces X . A natural question that comes with this problem, and which seems to be interesting enough to warrant special attention, is what spaces admit an open (continuous) mapping onto the real line R? Even a surface observation of continuous functions from reals to reals suggests that openness, and lack of local extrema, may be linked. However, the existence of a local extremum, considered as an obstruction to openness, is a local property; therefore it violates openness at a point, or in a neighbourhood of the point. So, in these circumstances, a localization of openness should be considered rather than the global notion. Further study of the topic leads to the conclusion that local connectedness of the domain space plays a role in solving the problem. A careful analysis of a variety of considered conditions indicates that connectedness im kleinen at a point is a more adequate property than local connectedness at this point. The obtained results, and unsolved problems related to the topic, are presented below. Let us mention that interiority of real functions has been a subject of interest to topologists for more than fifty years (see, e.g., [9]). Let topological spaces X and Y be given and let f : X → Y be a function from X into Y (not necessarily continuous). The function f is said to be interior at a point p ∈ X (see [8, p. 149]) provided that for each neighbourhood U of p we have f ( p) ∈ int f (U). The function f is said to be open if it maps open subsets of the domain X onto open subsets of the range Y . Obviously,