China
Africa
Explore chapters and articles related to this topic
Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
is called the discrete topology on X. Note that every set C containing x is its neighborhood in this topology. In particular every point is a neighborhood of itself. A totally opposite situation takes place if we define a topology by single set bases Bx={X} $ \mathcal B _x = \{ X \} $ , for every x∈X $ x \in X $ . Then, obviously, F(Bx)={X} $ \mathcal F ( \mathcal B _x) = \{ X \} $ and the only neighborhood of every x is the whole set X. The corresponding topology is known as the trivial topology on X. Notice that in the discrete topology every set is open, i.e., the family of open sets coincides with the whole P(X) $ \mathcal P (X) $ , whereas in the trivial topology the only two open sets are the empty set ∅ $ \emptyset $ and the whole space X. Obviously the trivial topology is the weakest topology on X while the discrete topology is the strongest one.
The divDiv-complex and applications to biharmonic equations
Published in Applicable Analysis, 2020
In this section, we consider bounded strong Lipschitz domains Ω of general topology and we will extend the results of the previous section as follows. The - and the -complexes remain closed and all associated cohomology groups are finite-dimensional. Moreover, the respective inverse operators are continuous and even compact, and corresponding Friedrichs/Poincaré type estimates hold. We will show this by verifying the compactness properties of Lemma 2.7 for the various linear operators of the complexes. Then Lemma 2.5, Remark 2.6, and Theorem 2.9 immediately lead to the desired results. Using Rellich's selection theorem, we have the following compact embeddings: The two missing compactness results that would immediately lead to the desired results are The main aim of this section is to show the compactness of the two crucial embeddings (22) and (23). As a first step we consider a trivial topology.