НЕКОТОРЫЕ СВОЙСТВА ПРОСТРАНСТВА τ-ЗАМКНУТЫХ ПОДМНОЖЕСТВ

This article investigates some properties of the space of τ-closed subsets of topological T 1 -spaces X.
In the family of τ-closed subsets, a base similar to the Vietoris topology is introduced. It is proven
that if X is a T 1 -space and X 0 ⊂ X, then the set {F : F ∈ exp τ X, X 0 ⊂ F} is closed in the space
exp τ X. It is also proven that if X is a T 1 -space, then the space exp τ X is also a T 1 -space. It is shown
that if Y is the everywhere τ-dense subset T 1 of the X space, then exp τ Y is the everywhere exp τ X
space. Similarly, if exp τ Y is everywhere dense in the space exp τ X, then it is shown that the space
Y is everywhere τ-dense in X. The analogy of Michael’s theorem for the density of the T 1 -space is
proven, i.e., let X be an infinite T 1 -space, then d τ (X) ≤ d τ (exp τ X). It is shown that if X is an
infinite T 1 -space and U is a τ-open subset, then any τ-open subset in U is a τ-open subset in X.