A topology generated by selections

Selections generating new topologies

“…In the present paper, we demonstrate that, for a second-countable zerodimensional space X, this topology may fail to be first-countable at some (or, even any) point of X. This settles some problems stated in [7]. …”

“…As usual, we write xEy to denote that (x, y) ∈ E. Let us recall that a relation E on X is anti-symmetric if xEy and yEx implies x = y. Following [7], we say that an anti-symmetric relation E on X is a selection relation if xEy or yEx for every x, y ∈ X. Let us emphasize that, in this terminology, a relation E on X is a linear order on X if E is a selection relation which is also transitive (i.e., xEy and yEz implies xEz).…”

“…Some further properties of this topology were studied in [4], [7]. For instance, by [7,Corollary 2.3], T f is always a regular topology on X.…”

“…For instance, by [7,Corollary 2.3], T f is always a regular topology on X. On the other hand, by [7,Corollary 2.4], T f is the usual Euclidean topology on the rational numbers Q, whenever f is a continuous weak selection for Q. Hence, it become quite natural to study this topology on the irrational numbers P which is an uncountable, second countable, zero-dimensional space.…”

“…Thus, in particular, there exists a continuous weak selection f for the irrational numbers P such that T f is not first-countable at any point of P (Corollary 4.2), which provides a negative answer to [7,Question 2], and a positive one to [7,Question 3]. Another interesting consequence is that an uncountable metrizable space X, with a covering dimension dim(X) = 0, is compact if and only if T f is second-countable for every continuous weak selection f on X (see Corollary 4.4).…”