Tietze Extension Does Not Always Work in Constructive Mathematics If Closed Sets Are Defined as Sequentially Closed Sets
Year: 2025
Type: article
Abstract: We prove that Tietze Extension does not always exist in constructive mathematics if closed sets on which the function we are extending are defined as sequentially closed sets. Firstly, we take a discrete metric space as our topological space. Now all sets open and sequentially closed. Then, we form an unextendible algorithmic function transforming positive integers to 0 and 1, looking at the preimages of these values as our sequentially closed sets. Then we show that if the Tietze theorem conclusion holds for these closed sets then the unextendible function is extendible thus giving us a contradiction. Hence, topology in constructive mathematics have great differences compared to standard topology on Euclidean space. In addition, different definition of special topological space may have converse result on the same theory. Hence, topology in constructive mathematics have great differences compared to standard topology on Euclidean space. In addition, different definition of special topological space may have converse result on the same theory.
Authors: Shun Ding, Yang Wan, Luofei Wang, Siqi Xiao
Institutions: University of Leeds, Nanjing Foreign Language School
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W4414723364