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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.
Elements of topology and homology
Published in Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama, Computational Topology for Biomedical Image and Data Analysis, 2019
Rodrigo Rojas Moraleda, Nektarios A. Valous, Wei Xiong, Niels Halama
Open cover A cover {Uα} of A in a topological space (X,T) is called open if all Uα are open in (X,T).
Roadmap to glory: scaffolding real analysis for deeper learning
Published in International Journal of Mathematical Education in Science and Technology, 2023
Timothy Huber, Josef Sifuentes, Aaron T. Wilson
Part (c) then brings us to our first delicate maneuver. An open cover of our domain must be established in a way that connects to the continuity criteria established in part (a). For each , there is an open set containing , and the family of these sets satisfies the definition of an open cover. This is the part that students will typically struggle with. Since lecture time is occupied with group work, we are free to intervene with any group that appears to be stuck in any place longer than is healthy. In such interventions we often remind students of the fundamentals of set theory that they would have seen in a previous introductory course to mathematical proofs, helping them to break this problem into smaller pieces of fundamental set theory. In part (d) we invoke the compactness of the function’s domain, and what this property implies about open covers. This step requires a solid grasp of the finite subcover property of the compactness condition, even if most of what is asked is a recitation of the definition of a subcover. These steps are big conceptual leaps for most students, and the scaffolding allows us to actively challenge the students to confront the steps with their working group.
Existence of positive solutions for a Schrödinger-Poisson system with critical growth
Published in Applicable Analysis, 2020
Next we shall state the existence of multiple positive solutions for (4) by Ljusternik-Schnirelmann category theory. We first recall some basic definitions. If Y is a closed subset of a topological space X, the Ljusternik-Schnirelmann category is the least number of closed and contractible sets in X which cover Y . In order to obtain multiple solutions for , we assume that and let In view of (or ), the set is compact. For any , we define the set . Our main results are as follows.
On nonlinear fractional Schrödinger equations with Hartree-type nonlinearity
Published in Applicable Analysis, 2018
A solution of (1.1) is called a ground state solution if its energy is minimal among the energy of all the nontrivial solutions of (1.1). We recall that, if Y is a closed subset of a topological space X, the Ljusternik–Schnirelmann category is the least number of closed and contractible sets in X which cover Y.