## You are here

Homevariable topology

## Primary tabs

# 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 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. 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.

## Mathematics Subject Classification

55U05*no label found*55U35

*no label found*55U40

*no label found*18G55

*no label found*18B40

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections