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Nolen Ryba
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I'm wondering if someone can furnish me with either an example of a topological space that is countable (cardinality) but not second countable or a proof that countable implies second countable. Thanks.
A countable but not second countable topological space is a mathematical concept that describes a topological space with a countably infinite number of open sets, but does not have a countable basis. This means that although there are infinitely many open sets, they cannot be written as a countable union of other open sets.
A second countable topological space is a topological space with a countable basis, meaning that the open sets can be written as a countable union of other open sets. In contrast, a countable but not second countable topological space does not have a countable basis, so the open sets cannot be written as a countable union of other open sets.
Yes, a countable but not second countable topological space can be compact. Compactness is a topological property that is independent of the basis or countability of the space. Therefore, a topological space can be both countable but not second countable and compact.
An example of a countable but not second countable topological space is the long line, which is a topological space that is formed by connecting infinitely many copies of the real line. The long line has a countably infinite number of open sets, but it does not have a countable basis.
Countable but not second countable topological spaces are important to study because they help to understand the limitations of countable bases in topological spaces. They also have applications in areas such as topology, analysis, and functional analysis. Additionally, studying these spaces can lead to new insights and discoveries in mathematics.