This analysis reveals the nature of Lebesgue covering dimension in continuous mappings, suggesting open questions for future exploration.
In Dimension Theory, there are “mapping theorems” that establish relationships between the dimensions of a domain and the range of a continuous mapping. Most of the theorems deal with mappings that satisfy additional conditions, such as being closed mappings. In the “environment” of such studies, the dimensions of continuous mappings have also been studied. In this paper, we introduce and investigate a new notion of dimension for continuous mappings between topological spaces, which is closer to the classical definition of the Lebesgue covering dimension of a space. We discuss various results concerning this dimension. Moreover, we present open questions and proposals of new dimensions of continuous mappings for further studies. They are based on known dimensions of continuous mappings between topological spaces.
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Georgiou et al. (2025) studied this question.
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