Closed graph property in Alexandroff spaces

Fatemah Ayatollah Zadeh Shirazi; Sajjad Moradi Chaleshtori

doi:10.56754/0719-0646.2803.429

Authors

Fatemah Ayatollah Zadeh Shirazi Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Enghelab Ave., Tehran, Iran. https://orcid.org/0000-0001-7735-5625
Sajjad Moradi Chaleshtori Faculty of Mathematics, Statistics and Computer Science, College of Science, University of Tehran, Enghelab Ave., Tehran, Iran. https://orcid.org/0009-0004-1167-2301

DOI:

https://doi.org/10.56754/0719-0646.2803.429

Keywords:

Alexandroff space, closed graph, connected component

Abstract

In the following text we show if \(X\) is an Alexandroff space, then \(f:X\to Y\) has closed graph if and only if it has constant closed value on each connected component of \(X\). Moreover, if \(X\) is an Alexandroff space and \(f:X\to Y\) has closed graph, then \(f:X\to Y\) is continuous. As a matter of fact, the number of maps which have closed graph from an Alexandroff space \(X\) to a topological space \(Y\) depends just on the number of connected components of \(X\) and the number of closed points of \(Y\).

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References

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