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Preliminaries
Published in Hugo D. Junghenn, Principles of Analysis, 2018
A topological space is said to be first countable if each point x has a countable neighborhood base. A metric space is first countable; for example, the collection of open (or closed) balls at x with radii 1 / n (n∈N $ n \in \mathbb N $ ) is a countable neighborhood base.
Topological and Metric Spaces
Published in J. Tinsley Oden, Leszek F. Demkowicz, Applied Functional Analysis, 2017
J. Tinsley Oden, Leszek F. Demkowicz
Sequential Topological Spaces. A topological space X is said to be sequential if one of the following equivalent conditions holds.Every set G⊂X $ G \subset X $ is open iff it is sequentially open.Every set F⊂X $ F \subset X $ is closed iff it is sequentially closed.The equivalence of the two conditions follows immediately from the duality principle that holds for both standard and sequentially open and closed sets. As we have learned in this section, every first countable topological space is sequential. The converse is not true. There exist sequential topological spaces that are not first countable. However, sequential topological spaces share many properties with the first countable spaces. For instance, if X is sequential then any sequentially continuous function f:X→Y $ f:X \rightarrow Y $ , mapping X into any topological space Y, is automatically continuous; see Exercise 4.4.5. In fact, the condition, if satisfied for every function f (and topological space Y), implies that X must be sequential; see Exercise 4.4.6. In a sense, the sequential topological spaces represent the largest class of spaces where topological properties like openness, closedness, and continuity may be equivalently formulated with sequences.
On the Completion of Fuzzy Normed Linear Spaces in the Sense of Bag and Samanta
Published in Fuzzy Information and Engineering, 2021
A fuzzy set U on X is called Q-neighbourhood of iff there exists such that .A family of Q-neighbourhoods of is called Q-neighbourhood base of iff for every Q-neighbourhood A of , there exists such that . is called first countable, if for each , there exists Q-neighbourhood base of such that has countable fuzzy sets.