Question

Define Dense subset .​

Answer 1

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A constitutes the whole set X. The density of a topological space X is the least cardinality of a dense subset of X

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Answer 2

In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A; that is, the closure of A constitutes the whole set X. ... The density of a topological space X is the least cardinality of a dense subset of X

Theorem

There exists a dense subset ℋ of such that for every h ∈ ℋ, Equation (23) displays chaotic dynamics. Moreover, for all is open in .To prove this theorem, one first refers to Theorem 10 to prove the result in a simplified setting. If is a true minimizer for Ah, n over if .

Corollary 3.5

Let hypotheses (H1)-(H3), (N1)-(N3) be satisfied with m ⩾ 1. Let D be an open set in Xα such that

Then there exist an open and dense subset G ⊂ D and a positive integer q0 such that for any u0 ∈ G there is a solution ϕ(x, t) of (1.1) and (1.2) with the following properties:

(i)

ϕ(x, t) is q τ-periodic in t for some q ⩽ q0,

(ii)

ϕ(·, 0) is at least neutrally stable as a periodic point of the period map ,

(iii)

if f is independent of t then so is ϕ (hence it is an equilibrium).If hypotheses (G1)-(G3) are satisfied then (B) can be omitted.

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