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Topological Compactness
Published in James K. Peterson, Basic Analysis I, 2020
Example 5.1.1Unions need not be over countable index sets. A good example is this. LetS=[1,3)which has an infinite number of points. At each x ∈ S, pick the radius rx > 0 and consider the setA=∪x∈SBrx. This index set is not an RII so indeed is an example of what is called a uncountable set. To be precise, acountableset Ω is one for which there is a map f which assigns each element in Ω one and only one element of ℕ. A RII is countable because nk is assigned to k in ℕ and this assignment is unique. Hence there is a 1 – 1 and onto map from the RII to ℕ.
Introduction and Mathematical Preliminaries
Published in L. Prasad, S. S. Iyengar, WAVELET ANALYSIS with Applications to IMAGE PROCESSING, 2020
The cardinality of R is called the power of the continuum, denoted ℵ1. The continuum hypothesis of Cantor states that there are no infinite sets whose cardinality is less than the power of the continuum and greater than that of a countable set.
On analysis and design of discrete-time constrained switched systems*
Published in International Journal of Control, 2018
Matheus Souza, André R. Fioravanti, Robert N. Shorten
For real matrices and vectors, () indicates transpose. For square matrices, tr( · ) denotes the trace function. The sets of natural, real, and nonnegative real numbers are indicated by , and , respectively. For any countable set , its cardinality is denoted by . Usual norms adopted in this paper are denoted as follows; the Euclidean norm of a vector x in is denoted by ; the norm for a continuous-time signal x is denoted by ; similarly, the ℓ2 norm for a discrete-time signal x is denoted by . For symmetric matrices, the symbol (⋆) denotes each of its symmetric blocks. We also define two important subsets of Metzler matrices: consists of all matrices such that πji ≥ 0, for all i ≠ j, and ∑Nj = 1πji = 0, for i = 1,… , N; consists of all matrices such that πji ≥ 0, for all i, j, and ∑Nj = 1πji = 1, for i = 1,… , N. Finally, X > 0 (X ≥ 0) denotes that the symmetric matrix X is positive definite (positive semidefinite); the set of all (positive definite) symmetric matrices of order n is denoted by () .