# 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 $\mathrm{\Lambda}$. 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 $\lambda \in \mathrm{\Lambda}$
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^{} ${\U0001d5a3}_{i}$
where the vertical arrows denote the assignment of topologies
${\mathcal{T}}_{i}$ to the class of objects of the ${\U0001d5a3}_{i}$ along
with functors^{} ${\mathcal{F}}_{i}:{\U0001d5a3}_{i}\u27f6{\U0001d5a3}_{i+1}$, 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 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 |