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 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. 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.