basic constructions:
strong axioms
further
topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
The term mereology was coined by the logician Stanislaw Lesniewski from greek ‘meros’-part to designate the study of the parthood relation.
Traditionally a ‘part’ of philosophy, the modern mathematical theory started with the work of the Polish school of logic in the early 20th century in the context of the foundations of mathematics. These early approaches went hand in hand with systems of pointless topology.
More recently, the main impact for the field comes from computer science in the context of logics for spatial reasoning and computer vision.
With the inception of topos theory William Lawvere has conceived of logic from the perspective of mereology by making the role of the subobject classifier manifest.
More specifically, he has pointed out in the 1980s the relevance of co-Heyting and bi-Heyting structures to mereology in this context and proposed to view logical theories from the point of view of a ‘mereodynamics’.
P. Forrest, Nonclassical Mereology and Its Application to Sets , Notre Dame J. Form. Logic 43 no.2 (2003) pp.79-94. (pdf)
C. Kuratowski, Sur l’opération $\bar{A}$ de l’analysis situs , Fund. Math. III (1922) pp.192-195. (pdf)
F. W. Lawvere, Introduction , pp.1-16 in Lawvere, Schanuel (eds.), Categories in Continuum Physics , Springer LNM 1174 1986.
F. W. Lawvere, Intrinsic Co-Heyting Boundaries and the Leibniz Rule in Certain Toposes , pp.279-281 in Carboni, Pedicchio, Rosolini (eds.) , Category Theory Como Conference, Springer LNM 1488 (1991).
S. Lesniewski, Grundzüge eines neuen Systems der Grundlagen der Mathematik , Fund. Math. XIV pp.1-81 (1929).^{1} (pdf)
T. Mormann, Heyting Mereology as a Framework for Spatial Reasoning , Axiomathes 23 no.1 (2013) pp.237-264. (draft)
W. Noll, The Geometry of Contact, Separation and Reformation of Continuous Bodies , Research report no.92-NA-029 Carnegie Mellon University 1992. (pdf)
W. Noll, The geometry of contact, separation and reformation of continuous bodies , Arch. Rat. Mech. Anal. 122 (1993) pp.97-112.
Dana S. Scott, Geometry without Points , Talk Vienna 2014. (slides)
J.G. Stell, M.F. Worboys, The algebraic structure of sets of regions , pp.163-174 in Hirtle, Frank (eds.), Spatial Information Theory, Springer LNCS 1329 (1997).
Caveat: Lesniewski conceives his foundations as layered into a Protothetik , roughly the deductive system or ‘logistics’, an ontology , roughly the theory of classes, and, mereology. This paper that is one in row that stretches from 1927 to 1938 is concerned with the first two layers. ↩
Last revised on June 6, 2017 at 14:10:10. See the history of this page for a list of all contributions to it.