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Open Sets and Open Covers
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
Suppose S⊆R. Then a collection F of open sets is said to be an open cover for S if and only if the union of all open sets in F contains the set S. That is, F is an open cover for S if and only if S⊆⋃O∈FO. If F is an open cover for a set S, we will often say that F “covers” S.
Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A collection U $ \boldsymbol{{ \fancyscript {U}}} $ of open subsets of a topological space X such that ⋃U=X $ \bigcup \boldsymbol{{ \fancyscript {U}}}= X $ is called an open cover of X . A subcollection of U $ \boldsymbol{{ \fancyscript {U}}} $ that is a cover of X is called a subcover. A space X is said to be compact if each open cover of X has a finite subcover. A subset Y of X is compact if it is compact in the relative topology, that is, if for each collection U $ \boldsymbol{{ \fancyscript {U}}} $ of open sets in X there exists a finite subcollection U0 $ \boldsymbol{{ \fancyscript {U}}}_0 $ such that Y⊆⋃U0 $ Y \subseteq \bigcup \boldsymbol{{ \fancyscript {U}}}_0 $ . A subset Y of X is relatively compact if its closure is compact.
Appendices
Published in Craig A. Tovey, Linear Optimization and Duality, 2020
Compactness is often phrased as “every open cover contains a finite subcover.” Fortunately, we can use a much simpler definition in ℜn with the Euclidean (or sup or L1) metric.
On the topological entropy of induced transformations for free semigroup action
Published in Dynamical Systems, 2021
Next, we define the topological entropy of an open cover for free semigroup actions. Proposition 2.2 reveals that the topological entropy of can be defined by using open covers. Let be an open cover of X, i.e., a family of open subsets of X whose union is X. Set . For any two open covers and , set . Denote by the minimal cardinality of all subcovers of . For any , set and .