variable topology

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

,

𝒯

)

where

𝒯

is a
topology on

X

. For

instance

, suppose an

open set

topology

is given by the set

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

𝒰

λ

of

a
system of prescribed open sets

, where

λ

belongs to some

indexing set

Λ

. 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

,

𝒯

λ

)

where

λ

∈

Λ

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

𝖣

i

where the vertical arrows denote the assignment of topologies

𝒯

i

to the class of objects of the

𝖣

i

along
with

functors

ℱ

i

:

𝖣

i

⟶

𝖣

i

+

1

, for

1

≤

i

≤

n

-

1

:

\diagram

&

𝒯

1

\dto

<

-

.05

e

x

>

&

𝒯

2

\dto

<

-

1.2

e

x

>

&

⋯

&

𝒯

n

-

1

\dto

<

-

.05

e

x

>

&

𝒯

n

\dto

<

-

1

e

x

>

(

0.45

)

&

𝖣

1

\rto

ℱ

1

&

𝖣

2

\rto

ℱ

2

&

&

⋯

\rto

ℱ

n

-

1

𝖣

n

-

1

&

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

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