“School of Mathematics”
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| Paper IPM / M / 18024 |
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| Abstract: | |
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We introduce a new notion of connectedness in topological spaces that diverges from the traditional definition. In this framework, a topological space is considered ``connected'' if it cannot be divided into two ``large'' open subsets. The notion of a filter, representing a collection of ``large'' subsets, or its dual notion, an ideal, which represents ``small'' subsets, naturally emerges in this context.
Formally, a topological space is said to be connected modulo an ideal of its subsets if there does not exist a pair of disjoint open subsets whose union is the entire space, with neither belonging to the ideal. We refer to this new concept as ``weak connectedness modulo an ideal,'' distinguishing it from the existing, stronger (as we will demonstrate) notion of connectedness modulo an ideal. We explore the relationship between these two concepts and demonstrate how standard results on connectedness have analogous counterparts within this generalized framework. Special focus is given to completely regular spaces. Our approach expands the understanding of connectedness in topology and lays the groundwork for further research into the interaction between ideals and topological properties.
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