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Journey into special T1-spaces

  • Robert Rałowski
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Abstract

In this survey, we review certain types of special T1 spaces and their associated fixed-point theorems. Kupka introduced the notion of a feeble topological contraction, which naturally generalizes Lipschitz contractions defined on metric spaces. Specifically, Kupka established a fixed-point theorem for feeble topological contractions possessing a closed graph within the product of arbitrary T0 spaces. Furthermore, the T1 separation axiom is shown to guaranty the uniqueness of such fixed points. Finally, we discuss peripheral Hausdorff and locally Hausdorff spaces within the context of these fixed-point results.

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Cite

BibTeXRIS
BibTeX
@misc{ralowski2026journey,
  title = {{Journey into special $T_1$-spaces}},
  author = {Robert Rałowski},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.20602},
  eprint = {2609.20602},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.20602},
}
RIS
TY  - UNPB
TI  - Journey into special T_1-spaces
AU  - Robert Rałowski
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.20602
UR  - https://arxiv.org/abs/2609.20602
AB  - In this survey, we review certain types of special T_1 spaces and their associated fixed-point theorems. Kupka introduced the notion of a feeble topological contraction, which naturally generalizes Lipschitz contractions defined on metric spaces. Specifically, Kupka established a fixed-point theorem for feeble topological contractions possessing a closed graph within the product of arbitrary T_0 spaces. Furthermore, the T_1 separation axiom is shown to guaranty the uniqueness of such fixed points. Finally, we discuss peripheral Hausdorff and locally Hausdorff spaces within the context of these fixed-point results.
ER  -