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Zenodo
2025
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| Accesso online: | https://doi.org/10.5281/zenodo.17536504 |
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| _version_ | 1866901617645715456 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | This paper presents a theoretical exploration into the unification of connectivity and compactness, two fundamental concepts in general topology. While traditionally treated as distinct properties, we argue that they are deeply intertwined aspects of a topological space's global structure, representing complementary notions of wholeness and boundedness. Connectivity provides a notion of indivisibility, ensuring a space cannot be broken into separate open pieces. Compactness provides a form of topological finiteness, ensuring that any open cover can be reduced to a finite subcover. This work synthesizes established topological results to build a conceptual bridge between them. We first analyze their interdependent roles in key theorems, such as the properties of continuous maps on compact and connected domains. The methodology then shifts to more abstract frameworks where their unification becomes more explicit. We investigate the category of compactly generated spaces, where the topology itself is determined by its compact subspaces, thereby making the global property of connectivity intrinsically dependent on the 'finite' nature of compactness. Furthermore, we explore a higher-level unification through the lens of category theory, framing compactness and connectivity as manifestations of dual principles governing how a space relates to its parts and its environment. The results of this synthesis suggest that connectivity can be viewed as an internal coherence, while compactness is a form of external constraint or self-containment. The discussion posits that this unified perspective offers a more profound understanding of topological invariants and has implications for fields such as algebraic topology and topological data analysis, where the interplay between local 'finite' data and global 'whole' structure is paramount. We conclude that viewing compactness and connectivity not as separate axioms but as a unified pair of principles offers a more robust and insightful foundation for geometric and topological reasoning. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17536504 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Unification of Connectivity and Compactness SÉRGIO DE ANDRADE, PAULO This paper presents a theoretical exploration into the unification of connectivity and compactness, two fundamental concepts in general topology. While traditionally treated as distinct properties, we argue that they are deeply intertwined aspects of a topological space's global structure, representing complementary notions of wholeness and boundedness. Connectivity provides a notion of indivisibility, ensuring a space cannot be broken into separate open pieces. Compactness provides a form of topological finiteness, ensuring that any open cover can be reduced to a finite subcover. This work synthesizes established topological results to build a conceptual bridge between them. We first analyze their interdependent roles in key theorems, such as the properties of continuous maps on compact and connected domains. The methodology then shifts to more abstract frameworks where their unification becomes more explicit. We investigate the category of compactly generated spaces, where the topology itself is determined by its compact subspaces, thereby making the global property of connectivity intrinsically dependent on the 'finite' nature of compactness. Furthermore, we explore a higher-level unification through the lens of category theory, framing compactness and connectivity as manifestations of dual principles governing how a space relates to its parts and its environment. The results of this synthesis suggest that connectivity can be viewed as an internal coherence, while compactness is a form of external constraint or self-containment. The discussion posits that this unified perspective offers a more profound understanding of topological invariants and has implications for fields such as algebraic topology and topological data analysis, where the interplay between local 'finite' data and global 'whole' structure is paramount. We conclude that viewing compactness and connectivity not as separate axioms but as a unified pair of principles offers a more robust and insightful foundation for geometric and topological reasoning. |
| title | The Unification of Connectivity and Compactness |
| url | https://doi.org/10.5281/zenodo.17536504 |