PlanetPhysics/Variable Topology

Key data
Let us recall the basic notion that a \htmladdnormallink{topological {http://planetphysics.us/encyclopedia/CoIntersections.html} space} 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 sets, a compact-open topology, and so on (see Brown, 2006). In any case, a topological space consists of a pair $$(X, \mathcal T)$$ where $$\mathcal T$$ is a topology on $$X$$. For instance, suppose an open set topology is given by the set $$\mathcal U$$ of prescribed open sets of $$X$$ satisfying the usual axioms (Brown, 2006 Chapter 2).

Definition of Variable Topology
Now, to speak of a variable open-set topology one might conveniently take in this case a family of sets $$\mathcal U_{\lambda}$$ of \emph{a system of prescribed open sets}, where $$\lambda$$ belongs to some indexing set $$\Lambda$$. 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 $$(X, \mathcal T_{\lambda})$$ where $$\lambda \in \Lambda$$ is an index set.

Examples A straightforward example of a network system with variable topology is that of a family of graphs generated over a fixed set of vertices by changing the graph edges or connections between its vertices.

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 categories $$\mathsf D_i$$ where the vertical arrows denote the assignment of topologies $$\mathcal T_i$$ to the class of objects of the $$\mathsf D_i$$ along with functors $$\F_{i} : \mathsf D_{i} \lra \mathsf D_{i+1}$$, for $$1 \leq i \leq n-1$$~:

$$ \diagram &  \mathcal T_{1} \dto<-.05ex> & \mathcal T_{2} \dto<-1.2ex> & \cdots & \mathcal T_{n-1} \dto <-.05ex> &  \mathcal T_{n} \dto<-1ex>_(0.45){} \\ & \mathsf D_{1}\rto^{\F_1} & \mathsf D_{2} \rto^{\F_2}  \rule{0.5em}{0ex}  & & \cdots \rto^{\F_{n-1}} \rule{0.5em}{0ex} \mathsf D_{n-1} & \rule{0em}{0ex} \mathsf D_{n} \enddiagram $$

In this way a variable topology 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 \htmladdnormallink{groupoid {http://planetphysics.us/encyclopedia/QuantumOperatorAlgebra5.html} 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), symmetry groupoids of various cell networks would appear relevant to the physiology of animal locomotion as one example.