Switching to a coarser topology preserves countable compactness. This property of topological spaces is preserved under coarsening, viz, if a set with a given topology has the property, the same set with a coarser topology also has the property Theorem 4.2 Let H be a Hilbert space and M a closed subspace of H. View all weakly hereditary properties of topological spaces | View all subspace-hereditary properties of topological spacesĪny closed subset of a countably compact space is countably compact, when endowed with the subspace topology. see that if the space is finite dimensional all the norms defined on it. Let be any closed immersion of locally ringed spaces. Clearly, this is (isomorphic) to the closed subspace associated to the quasi-coherent sheaf of ideals, as in Example 26.4.3. Weak hereditariness This property of topological spaces is weakly hereditary or closed subspace-closed in other words, any closed subset (equipped with the subspace topology) of a space with the property, also has the property. Thus we see that is a closed immersion of locally ringed spaces, see Definition 26.4.1. The proof of this follows from a version of the tube lemma. View all properties of topological spaces closed under products (a) Show that if A is closed in Y and Y is closed in X, then A is. Metaproperties Products This property of topological spaces is closed under taking arbitrary products Let Y X, and give Y the subspace topology. linear subspace) of V iff W, viewed with the operations it inherits from V, is itself a vector space. (infact GT is a Banach space as a subspace of X × Y ).
Relation with other properties Stronger properties Graph GT being a closed set in X ×Y, it follows that GT is complete. It is also an instance of the countably qualifier applied to compactness-like properties. Viz, every countable open cover has a finite open refinement. In the refinement formalism, a refinement formal expression is: P : H H such that P X and each element x can be written unqiuely as a sum a + b. Every point-finite open cover has a finite subcover.įurther information: equivalence of definitions of countably compact space Formalisms Refinement formal expression If X is any closed subspace of H then there is a bounded linear operator.In other words, given a countable collection of open subsets whose union is the whole space, there is a finite subcollection whose union is again the whole space.
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