Preliminary data Let us recall the basic notion that a topological space consists of a set and a ‘topology’ on 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 sets, a compact-open topology, and so on (see Brown, 2006). In any case, a topological space consists of a pair where is a topology on . For instance, suppose an open set topology is given by the set of prescribed open sets of 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 of a system of prescribed open sets, where belongs to some indexing set . 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).
In general, we may speak of a topological space with a varying topology as a pair where is an index set.
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 -stage dynamic evolution (through complexity) of categories where the vertical arrows denote the assignment of topologies to the class of objects of the along with functors , for :
In this way a variable topology (http://planetmath.org/VariableTopology) can be realized through such -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 (http://planetmath.org/VariableTopology3). As suggested by Golubitsky and Stewart (2006), symmetry groupoids of various cell networks would appear relevant to the physiology of animal locomotion as one example.
|Date of creation||2013-03-22 18:15:39|
|Last modified on||2013-03-22 18:15:39|
|Last modified by||bci1 (20947)|
|Defines||indexed family of topological spaces|