<P> For any base set, , an arbitrary mapping of subsets to subsets can be called a pseudoclosure (pseudointerior) function, , with its dual of a pseudointerior (pseudoclosure), , related by the involution in dssmapfvd. As gains properties of the closure (interior) function of a topology on , so does its dual gain corresponding properties of the interior (closure) function of that topology. </P> <P> As there is also a natural isomorphism which maps from to (and likewise for and , introduced below) which identically gains the properties of the neighborhood function of a topology (modified and restricted to operate on single points). A function dual to , which Stadler and Stadler refer to as a convergent function, is represented by in this section. </P> <!-- TODO: evaluate replacing 'convergent function' with 'adherents function' and replace 'neighborhood function' with 'neighborhoods function' so that we might more clearly refer to the members of the value of the functions as 'an adherent' and 'a neighborhood'. Likewise the collection N(x) could be a 'neighborhood system' and by analogy M(x) would be an 'adherent system'. BJ suggested 'adherent' but I want to think about it. --> <P> Based on this and the early treatment of topology in Seifert and Threlfall, it seems reasonable to define a pseudotopology as defined in terms of its base set and one of these functions with theorems treating the equivalence of the other definitions and adding topological structure if enough properties hold true. </P> <TABLE> <TR> <TD WIDTH="15%"> </TD> <TH WIDTH="17%"> Neighborhoods </TH> <TH></TH> <TH WIDTH="17%"> Interior </TH> <TH></TH> <TH WIDTH="17%"> Closure </TH> <TH></TH> <TH WIDTH="17%"> Convergents </TH> <TH> Theorems </TH> </TR> <TR> <TH WIDTH="15%"> Functions </TH> <TD WIDTH="17%" STYLE="text-align: center;"> </TD> <TD></TD> <TD WIDTH="17%" STYLE="text-align: center;"> </TD> <TD></TD> <TD WIDTH="17%" STYLE="text-align: center;"> </TD> <TD></TD> <TD WIDTH="17%" STYLE="text-align: center;"> </TD> <TD> </TD> </TR> <TR> <TD ROWSPAN="2" STYLE="text-align: center;"> <DIV STYLE="font-weight: bold"> Correspondences </DIV> (assuming ) </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD ROWSPAN="2"> ntrclselnel1, ntrneiel, neicvgel1 </TD> </TR> <TR> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> </TR> <TR> <TH> Neighborhoods </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrneifv3, clsneifv3, neicvgfv </TD> </TR> <TR> <TH> Interior </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrneifv4, ntrclsfv, clsneifv4 </TD> </TR> <TR> <TH> Closure </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> clsneifv4, ntrclsfv, ntrneifv4 </TD> </TR> <TR> <TH> Convergents </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> neicvgfv, clsneifv3, ntrneifv3 </TD> </TR> </TABLE> <P> We have the following table of equivalences to axioms largely established by Kuratowski. In the formulas in this table, to reduce the width of the columns, if any of the variables , , or are used, then they are implicitly universally quantified and (respectively and ) ranges over (respectively and ). </P> <!-- The following table adapted from: Bärbel M. R. Stadler, Peter F. Stadler. "Basic Properties of Closure Spaces." https://www.academia.edu/12565065/Basic_Properties_of_Closure_Spaces with variants also in: Bärbel M. R. Stadler, Peter F. Stadler. "Higher Separation Axioms in Generalized Closure Spaces." Comment. Math. Warszawa, Ser. I, 43: 257-273, 2003. Preprint: https://www.tbi.univie.ac.at/papers/Abstracts/01-pfs-017.pdf Stadler claims that K3 implies KB, but counterexamples exist. A pseudo-closure function of ( x e. ~P B |-> ( B \ x ) ), for nonempty base set, satisfies K0' and K3 but not KB. See clsk3nimkb for proof of this counterexample. Likewise, K0 is unnecessary to demonstrate KB implies KA. See ntrkbimka . --> <TABLE> <TR> <TD WIDTH="15%"> Assuming a prefix of: <BR> </TD> <TH WIDTH="17%"> Neighborhoods </TH> <TH WIDTH="17%"> Interior </TH> <TH WIDTH="17%"> Closure </TH> <TH WIDTH="17%"> Convergents </TH> <TH> Equivalence Theorems </TH> </TR> <TR> <TH> K0' <BR> Neighborhoods are nonempty. </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclsneine0, ntrneineine0, ntrneineine1 </TD> </TR> <TR> <TH> KA' <!-- invented name for the dual of K0' --> <BR> No neighborhood is equal to the full powerset. </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclsneine0, ntrneineine0, ntrneineine1 </TD> </TR> <TR> <TH> K0 <BR> Preservation of the Nullary Union of Closures </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclscls00, ntrneicls00, ntrneicls11 </TD> </TR> <TR> <TH> KA <BR> Preservation of the Nullary Union of Interiors </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclscls00, ntrneicls00, ntrneicls11 </TD> </TR> <TR> <TH> K1 <BR> Isotonic <BR> Montonic </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center; line-height: 1em;"> <BR> <SPAN STYLE="color: #AAAAAA;"> — or — </SPAN> <BR> <BR> <SPAN STYLE="color: #AAAAAA;"> — or — </SPAN> <BR> </TD> <TD STYLE="text-align: center; line-height: 1em;"> <BR> <SPAN STYLE="color: #AAAAAA;"> — or — </SPAN> <BR> <BR> <SPAN STYLE="color: #AAAAAA;"> — or — </SPAN> <BR> </TD> <TD STYLE="text-align: center;"> </TD> <TD> isotone1, isotone2, ntrclsiso, ntrneiiso </TD> </TR> <TR> <TH> K2 <BR> Closure is Expansive </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclsk2, ntrneik2, ntrneix2 </TD> </TR> <TR> <TH> KB <BR> Non-disjoint Neighborhoods </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclskb, ntrneikb, ntrneixb </TD> </TR> <TR> <TH> K3 <BR> Closure is Sub-linear </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclsk3, ntrneik3, ntrneix3 </TD> </TR> <TR> <TH> K13 <BR> Closure is finitely linear </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclsk13, ntrneik13, ntrneix13 </TD> </TR> <TR> <TH> K4 <BR> Closure is idempotent </TH> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD STYLE="text-align: center;"> </TD> <TD> ntrclsk4, ntrneik4 </TD> </TR> <!-- TODO: K5, XF = Every neighborhood is a Filter K13 <-> K1 + K3 --> </TABLE> <P> Using these properties as axiomic constraints on the functions, certain collections of them give rise to named spaces. </P> <!-- The following table adapted from: Bärbel M. R. Stadler, Peter F. Stadler. "Basic Properties of Closure Spaces." https://www.academia.edu/12565065/Basic_Properties_of_Closure_Spaces with variants also in: Bärbel M. R. Stadler, Peter F. Stadler. "Generalized Topological Spaces in Evolutionary Theory and Combinatorial Chemistry." J. Chem. Inf. Comput. Sci. 2002, 42, 3, 577-585 https://doi.org/10.1021/ci0100898--> <TABLE> <TR> <TH>Space</TH> <TH>Foundational Axioms</TH> <TH>Derived Axioms</TH> <TH>Theorems</TH> </TR> <TR> <!-- Á. Csázár. "Generalized topology, Generalized continuity." Acta. Math. Hungar. 96: 351-357, 2002. --> <TD> Csázár Generalized Neighborhood Space </TD> <TD> K2 </TD> <TD> KA', KA, KB </TD> <TD> ntrk2imkb, ntrkbimka, neik0imk0p </TD> </TR> <TR> <!-- Won Keun Min. "Results on Strong Generalized Neighborhood Spaces." J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 15(3): 221-227, August 2008. --> <TD> <!-- Won Keun --> Min Strong Generalized Neighborhood Space </TD> <TD> K2, K3 </TD> <TD> KA', KA, KB </TD> <TD> ntrk2imkb, ntrkbimka, neik0imk0p </TD> </TR> <TR> <!-- S. Gniłka. "On extended topologies. I: Closure operators." Ann. Soc. Math. Pol. Ser. I, Commentat. Math. 34:81-94, 1994. S. Gniłka. "On extended topologies. II: Compactness, quasi-metrizability, symmetry." Ann. Soc. Math. Pol. Ser. I, Commentat. Math. 35:99-108, 1997. --> <TD> Gniłka Extended Topology</TD> <TD> K0', K1 </TD> <TD> K0 </TD> <TD> neik0pk1imk0 </TD> </TR> <TR> <!-- M. M. Brissaud. "Les espaces prétopologiques." C. R. Acad. Sc. Paris Ser. A 280:705-708, 1975. --> <TD> Brissaud Space </TD> <TD> K0, K2 </TD> <TD> K0', KA', KA, KB </TD> <TD> neik0imk0p, ntrk2imkb, ntrkbimka </TD> </TR> <TR> <TD> Neighborhood Space </TD> <TD> K0', K1, K2 </TD> <TD> K0, KA', KA, KB </TD> <TD> neik0pk1imk0, ntrk2imkb, ntrkbimka, neik0imk0p </TD> </TR> <TR> <!-- Appears to be a reference to B. A. Davey, H. A. Priestley. "Introduction to Lattices and Order." Cambridge University Press, 2002. p. 48. where the pseudo-closure is: ( s e. ~P B |-> |^| { t e. C | s C_ t } ) where C is presumably a subset of ~P B such that every nonempty subset has its intersection in C . --> <TD> Davey and Priestley Intersection Structure </TD> <TD> K1, K4 </TD> <TD> </TD> <TD> </TD> </TR> <TR> <TD> Moore Closure Space </TD> <TD> K1, K2, K4 </TD> <TD> KA', KA, KB </TD> <TD> ntrk2imkb, ntrkbimka, neik0imk0p </TD> </TR> <TR> <!-- W. P. Soltan. "An Introduction in Axiomatic Theory of Convexity." Shtiintsa, Kishinev, 1984. Russian. --> <TD> Convex Closure Space </TD> <TD> K0', K1, K2, K4 </TD> <TD> K0, KA', KA, KB </TD> <TD> neik0pk1imk0, ntrk2imkb, ntrkbimka, neik0imk0p </TD> </TR> <TR> <!-- M. B. Smyth. "Semi-metric, closure spaces and digital topology." Theor. Computer Sci. 151: 257-276, 1995. --> <TD> Smyth Neighborhood Space </TD> <TD> K0', K13 </TD> <TD> K0, K1, K3 </TD> <TD> neik0pk1imk0, ntrk1k3eqk13 </TD> </TR> <TR> <TD> Čech Closure Space <BR> Pretopological Space </TD> <TD> K0', K2, K13 </TD> <TD> K0, K1, KA', KA, KB, K3 </TD> <TD> neik0pk1imk0, ntrk2imkb, ntrkbimka, neik0imk0p, ntrk1k3eqk13 </TD> </TR> <TR> <TD> Topological Space </TD> <TD> K0', K2, K13, K4 </TD> <TD> K0, K1, KA', KA, KB, K3 </TD> <TD> neik0pk1imk0, ntrk2imkb, ntrkbimka, neik0imk0p, ntrk1k3eqk13 </TD> </TR> <TR> <!-- P. Alexandroff. "Diskrete Räume." Math. Sb. (N.S.) 2: 501-518, 1937. R. E. Strong. "Finite Topological Spaces." Trans. Amer. Math. Soc. 123: 325-340, 1966. F. G. Arenas. "Alexandroff Spaces." Acta Math. Univ. Comenianae. 68: 17-25, 1999. --> <TD> Alexandroff Space </TD> <TD> K0', K2, K5 </TD> <TD> K0, K1, KA', KA, KB, K3, K13 </TD> <TD> neik0pk1imk0, ntrk2imkb, ntrkbimka, neik0imk0p, ntrk1k3eqk13, TBD <!-- AND K5 -> K1 + K3 --> </TD> </TR> <TR> <TD> Alexandroff Topological Space </TD> <TD> K0', K2, K4, K5 </TD> <TD> K0, K1, KA', KA, KB, K3, K13 </TD> <TD> neik0pk1imk0, ntrk2imkb, ntrkbimka, neik0imk0p, ntrk1k3eqk13, TBD </TD> </TR> </TABLE>