Compact subsets of metric space are closed. Thinking. 闭性不好搞,用补集的开性来证。 Proof. Let K be a compact subset of a metric space X. We shall prove that the complement of K is an open subset of X. Suppose p\in X, p\notin K. if q\in K, let V(p) and W(q) be neig...
quasi-metric spaceYoneda-completenessSmyth-completenessHausdorff quasi-metrichyperspace of nonempty compact subsets54B2054E99We study the hyperspace K 0 ( X ) of non-empty compact subsets of a Smyth-complete quasi-metric space ( X, d ). We show that K 0 ( X ), equipped with the Hausdorff ...
A thermodynamic definition of topological pressure for non-compact sets, ArtículoWe give a new definition of topological pressure for arbitrary (non-compact, non-invariant) Borel subsets of metric spaces. This new quantity is defined via a suitable variational principle, leading to an alternative ...
For a metric space (X, ρ), the following are equivalent: 1. X is compact. 2. Every collection of closed subsets of X with the finite intersection property has non-empty intersection. 3. X is sequentially compact. 4. X is complete and totally bounded. Proof. We’ll first show (...
We also show that any compact subset of a Hausdorff space X having a base of countable order must be a Gδ-subset of X and note that this result does not hold for countably compact subsets of BCO-spaces. We characterize CSS spaces in terms of certain functions g(n,x) and prove a ...
Closed Gδ-subsets of supercompact Hausdorff spaces Indag. Math., 41 (1979), pp. 155-162 View PDFView articleGoogle Scholar [19] C.F. Mills A simpler proof that compact metrizable spaces are supercompact Proc. AMS, 73 (1979), pp. 388-390 View in ScopusGoogle Scholar [20] C.F. ...
Then T is compact on Aαp(B) if and only if T fk → 0 in Aαp(B) for any bounded sequence { fk} in Aαp(B) such that fk → 0 uniformly on compact subsets of B. A proof can be found in [3, Proposition 3.11] for a single composition operator and it can be easily ...
We also show that any compact subset of a Hausdorff space X having a base of countable order must be a G δ-subset of X and note that this result does not hold for countably compact subsets of BCO-spaces. We characterize CSS spaces in terms of certain functions g ( n , x ) and ...
Show activity on this post. Recall that the Hausdorff distance (metric) is: A metric in the space of subsets of a compact set KK, defined as follows. Let X,Y⊂KX,Y⊂K and let Dx,yDx,y be the set of all numbers ρ(x,Y)ρ(x,Y) and ρ(y,X)ρ(y,X) where ...
Let Γ denote the metric space of nonempty compact convex subsets of R__n with Hausdorff topology. For topological spaces X and Y denote by C(X→Y) the space of continuous mappings h: X → Y. For a subset S ⊂ R__n let V be a (multivalued) vector field on S, i.e., ...