variable topology

Preliminary dataLet us recall the basic notion that a topological spaceconsists of a set $X$ and a ‘topology’ on $X$ where the lattergives a precise but general sense to the intuitive ideas of‘nearness’ and ‘continuity’. Thus the initial task is toaxiomatize the notion of ‘neighborhood’ and then consider atopology in terms of open or of closed sets, a compact-opentopology, and so on (see Brown, 2006). In any case, a topologicalspace consists of a pair $(X,𝒯)$ where $𝒯$ is atopology on $X$. For instance, suppose an open set topologyis given by the set $𝒰$ of prescribed open sets of $X$satisfying the usual axioms (Brown, 2006 Chapter 2). Now, to speakof a variable open-set topology one might conveniently take inthis case a family of sets ${𝒰}_{\lambda }$ of asystem of prescribed open sets, where $\lambda$ belongs to someindexing set $\mathrm{\Lambda }$. The system of open sets may of course bebased on a system of contained neighbourhoods of points where onesystem may have a different geometric property compared say toanother system (a system of disc-like neighbourhoods compared withthose of cylindrical-type).

Definition 0.1.

In general, we may speak of a topological space with avarying topology as a pair $(X,{𝒯}_{\lambda })$ where $\lambda \in \mathrm{\Lambda }$is an index set.

Example The idea of a varying topology has been introduced to describe possible topologicaldistinctions in bio-molecular organisms through stages ofdevelopment, evolution, neo-plasticity, etc. This is indicatedschematically in the diagram below where we have an $n$-stagedynamic evolution (through complexity) of categories ${𝖣}_i$where the vertical arrows denote the assignment of topologies${𝒯}_i$ to the class of objects of the ${𝖣}_i$ alongwith functors ${ℱ}_i:{𝖣}_i⟶{𝖣}_{i+1}$, for$1\le i\le n-1$ :

In this way a variable topology (http://planetmath.org/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 approachinvolving concepts such as the free groupoid over a graph(Brown, 2006). Thus a varying graph system clearly induces anaccompanying system of variable groupoids (http://planetmath.org/VariableTopology3). As suggested byGolubitsky and Stewart (2006), symmetry groupoids of various cellnetworks would appear relevant to the physiology of animal locomotion as one example.