variable topology

Preliminary data Let us recall the basic notion that a topological spaceMathworldPlanetmath consists of a set X and a ‘topology’ on X where the latter gives a precise but general sense to the intuitive ideas of ‘nearness’ and ‘continuity’. Thus the initial task is to axiomatize the notion of ‘neighborhood’ and then consider a topology in terms of open or of closed setsPlanetmathPlanetmath, a compact-open topologyMathworldPlanetmath, and so on (see Brown, 2006). In any case, a topological space consists of a pair (X,𝒯) where 𝒯 is a topology on X. For instance, suppose an open set topology is given by the set 𝒰 of prescribed open sets of X satisfying the usual axioms (Brown, 2006 Chapter 2). Now, to speak of a variable open-set topology one might conveniently take in this case a family of sets 𝒰λ of a system of prescribed open sets, where λ belongs to some indexing set Λ. The system of open sets may of course be based on a system of contained neighbourhoods of points where one system may have a different geometric property compared say to another system (a system of disc-like neighbourhoods compared with those of cylindrical-type).

Definition 0.1.

In general, we may speak of a topological space with a varying topology as a pair (X,𝒯λ) where λΛ is an index setMathworldPlanetmath.

Example The idea of a varying topology has been introduced to describe possible topological distinctions in bio-molecular organisms through stages of development, evolution, neo-plasticity, etc. This is indicated schematically in the diagram below where we have an n-stage dynamic evolution (through complexity) of categoriesMathworldPlanetmath 𝖣i where the vertical arrows denote the assignment of topologies 𝒯i to the class of objects of the 𝖣i along with functorsMathworldPlanetmath i:𝖣i𝖣i+1, for 1in-1 :

\diagram&𝒯1\dto<-.05ex>&𝒯2\dto<-1.2ex>&&𝒯n-1\dto<-.05ex>&𝒯n\dto<-1ex>(0.45)&𝖣1\rto1&𝖣2\rto2 &&\rton-1 𝖣n-1&𝖣n\enddiagram

In this way a variable topologyPlanetmathPlanetmath ( can be realized through such n-levels of complexity of the development of an organism.

Another example is that of cell/network topologies in a categorical approach involving concepts such as the free groupoid over a graph (Brown, 2006). Thus a varying graph system clearly induces an accompanying system of variable groupoids ( As suggested by Golubitsky and Stewart (2006), symmetryPlanetmathPlanetmath groupoidsPlanetmathPlanetmathPlanetmath of various cell networks would appear relevant to the physiology of animal locomotion as one example.

Title variable topology
Canonical name VariableTopology
Date of creation 2013-03-22 18:15:39
Last modified on 2013-03-22 18:15:39
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 11
Author bci1 (20947)
Entry type Definition
Classification msc 55U05
Classification msc 55U35
Classification msc 55U40
Classification msc 18G55
Classification msc 18B40
Synonym variable topology
Related topic TopologicalSpace
Related topic VariableTopology3
Defines indexed family of topological spaces