Variations of topologies induced by mixing topologies
Keywords:
Euclidean topology, Sorgenfrey line, Hattori spaces, almost topological groups, countable dense homogeniety, compact, $\sigma$-compact, $C_p(X)$, hybrid topologyAbstract
The article concerns a family of topologies defined by mixing topologies in spaces, which depends on subsets of spaces. In 2010, a family of topologies were defined by mixing the Euclidean topology and the Sorgenfrey line topology in the set $\mathbb{R}$ of real numbers ([19]). Since then, several authors have researched these topologies and their surroundings. The purpose of this article is to summarize the results on this topic obtained mostly in recent ten years.
Several topological properties, for example, the completeness, cardinal invariants, and the countable dense homogeneity are considered. The idea of inducing the family of topologies in [19] is applied to almost topological groups in [7]. We also have the Lindelöf property and the normality of the space of real valued continuous functions with the topology of pointwise convergence of these spaces. There are many variations of mixing topologies in spaces. We discuss topologies on $\mathbb{R}$ induced by mixing topologies other than the Euclidean and the Sorgenfrey topologies.
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