Related set . A set K, a subset of the line (or real numbers) is a related set if the closed interval formed by any of two of its points is also part of the set K.
Related set
In the general topology, one of the most important concepts is the related set, which is nothing more than the formalization of the intuitive idea that a figure or set, in consideration, is of a single piece.
In a way , topology , as a mathematical discipline , is the study of topological properties or invariants; among these: connectedness, compactness, interior, adherence, neighborhood, etc. With a somewhat elementary concept, you are going to define what a connected set is on the real line, without using the idea of an open set or separation.
We conceive of a related figure without fragmentation. One piece. But what is a connected set on the line like?
K is a connected set in R sss a, b ε K, implies [a; b] is part of K.
Examples
Any type of interval is a related set . They are also:
- I = <a; b> because if m and n are points of I, with m <n, it is true that a <m <n <b and every point x that is in [m; n] is also in I; therefore [m; n] is a subset of I.
- J = <a, b], since for [c; b], with a <c <b, we have that [c; b] is part of J.
- H = [a; b], since a and b are in H, it is verified that [a; b] is part of [a; b]
The set A = [1; 6] \ {3} because the closed interval [2; 4] is not part of [1; 6], even though 2 and 4 are in [1,6]. Well; A does not constitute, geometrically, a single piece.
Alternative definition
Let (R, T) be the usual topological space of R. A subset L of R is said to be connected at (R, T) if for every pair of open subsets C and D of (R, T), such that L is part of C union D, in addition L, C and D do not have common elements, it is inferred that L with C or L with D do not have common points.
Proposition
- A set L of real numbers (points on the real line) is a connected set of R if, and only if, it is an interval.
Corollary
Any range of real numbers is a related set
