VariableTopology 2009-04-19 15:01:57 2009-04-19 15:01:57 Topic network system with variable topology variable topological spaces varying topological spaces varying graph variable network variable quantum automaton variable topology varying topological spaces \textbf{Preliminary data} Let us recall the basic notion that a \emph{topological 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 \emph{open set topology} is given by the set $\mathcal U$ 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 $\mathcal U_{\lambda}$ 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). \begin{definition} In general, we may speak of a topological space with a \emph{varying topology} as a pair $(X, \mathcal T_{\lambda})$ where $\lambda \in \Lambda$ is an index set. \end{definition} \textbf{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 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 \PMlinkname{variable topology}{VariableTopology} 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 \emph{the free groupoid over a graph} (Brown, 2006). Thus a \emph{varying graph system} clearly induces an accompanying \PMlinkname{system of variable groupoids}{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.