# variable topology

Preliminary data Let us recall the basic notion that a 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 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 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).

###### Definition 0.1.

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.

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 $\mathcal{F}_{i}:\mathsf{D}_{i}{\longrightarrow}\mathsf{D}_{i+1}$, for $1\leq i\leq n-1$ :

 $\diagram&\mathcal{T}_{1}\dto<-.05ex>&\mathcal{T}_{2}\dto<-1.2ex>&\cdots&% \mathcal{T}_{n-1}\dto<-.05ex>&\mathcal{T}_{n}\dto<-1ex>_{(}0.45){}\\ &\mathsf{D}_{1}\rto^{\mathcal{F}_{1}}&\mathsf{D}_{2}\rto^{\mathcal{F}_{2}}% \rule{5.0pt}{0.0pt}&&\cdots\rto^{\mathcal{F}_{n-1}}\rule{5.0pt}{0.0pt}\mathsf{% D}_{n-1}&\rule{0.0pt}{0.0pt}\mathsf{D}_{n}\enddiagram$

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 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