Notes and reading materials on finite topological spaces

Notes written for the REU, and slides of a survey talk

• Finite topological spaces (pdf)
• Low dimensional examples (pdf)
• Finite spaces and simplicial complexes (pdf)
• An introduction to simplicial sets (JULY 21)
• (pdf)
• A few exercises and problems about finite topological spaces (UPDATED July 6) (pdf)
• Finite groups and finite spaces (pdf)
• Subdivisions of small categories (pdf)
• An outline summary of basic point set topology (pdf)
• Slides of a talk on finite topological spaces (pdf)

Selected papers and a thesis about finite topological spaces

• A. Fix and S. Patrias. Enumeration of homotopy classes of finite T_0 topological spaces. [REU paper] (pdf)
• J. Salvatore. The weak homotopy equivalence of S^n and a space with 2n+2 points. [REU paper] (pdf)
• E. Clader. Inverse limits of finite topological spaces [Undergraduate paper] (pdf)
• M. Kukiela. On homotopy types of Alexandroff spaces [Undergraduate paper] (pdf)
• J.A. Barmak (PhD thesis). Algebraic toplogy of finite topological spaces and applications (pdf)
• J.A. Barmak and E.G. Minian. Minimal finite models (pdf)
• T. Osaki. Reduction of finite topological spaces (pdf)
• J.A. Barmak and E.G. Minian. Simple homotopy types and finite spaces (pdf)
• J.A. Barmak and E.G. Minian. One-point reductions of finite spaces, h-regular CW-complexes and collapsibility (pdf)
• J.A. Barmak and E.G. Minian. Strong homotopy types, nerves, and collapses (pdf)
• J.A. Barmak and E.G. Minian. 2-Dimension from the topological viewpoint [combinatorics] (pdf)
• J.A. Barmak and E.G. Minian. Automorphism groups of finite posets [group theory] (pdf)
• J.A. Barmack. On Quillen's theorem A for posets [homotopy theory of posets] (pdf)
• K.A. Hardie, J.J.C. Vermeulen, and P.J. Witbooi. A nontrivial pairing of finite $T_0$-spaces (pdf)
• K.A. Hardie and P.J. Witbooi. Crown multiplications and a higher order Hopf construction (pdf)
• K.A. Hardie, S. Salbany, J.J.C. Vermeulen, and P.J. Witbooi. (pdf)
• J. Cupal, S. Kopp, P.F. Stadler. RNA shape space topology [biology!] (pdf)
• Rina Foygel. 2nd subdivision of a simplicial set (or category) (pdf)