variable topology
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).
Definition 0.1.
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.
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 |