Väljaspoole

Väljaspoole is a term used in Estonian mathematical literature to denote the exterior of a subset within a topological space. It is the part of the space that lies outside the set in question and is found by considering the openness of the surrounding area.

Formally, let X be a topological space with topology τ and A a subset of X. The väljaspoole of A, often denoted ext(A), can be defined in several equivalent ways: ext(A) = int(X \ A), i.e., the interior of the complement of A; ext(A) = X \ cl(A), i.e., the complement of the closure of A. Equivalently, est(A) consists of all points x in X for which there exists a neighborhood U of x with U ∩ A = ∅. In all these formulations, ext(A) is an open set and is disjoint from A.

Properties and relationships: ext(A) is contained in the complement of A’s closure, so it contains no limit points of A. The exterior is disjoint from A, and it forms part of the standard decomposition of X into ext(A), int(A), and ∂A (the boundary of A). The boundary ∂A is given by cl(A) ∩ cl(X \ A) and consists of points where every neighborhood meets both A and its complement; these points are not in ext(A).

Examples: In the real line with the usual topology, A = [0,1] has ext(A) = (-∞,0) ∪ (1,∞). If A is the set of rational numbers in R, ext(A) is empty, since the closure of A is all of R.

In summary, väljaspoole captures the open region outside a set, formalized as the interior of the space outside A or, equivalently, the complement of A’s closure.