Topology

From Maths
Revision as of 09:28, 30 December 2016 by Alec (Talk | contribs) (Added examples, see also, reference, minor fixes.)

(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Should be easy to flesh out, find some more references and demote to grade C once acceptable

Caution:This page is about topologies only, usually when talk of topologies we don't mean a topology but rather a topological space which is a topology with its underlying set. See that page for more details

Definition

A topology on a set [ilmath]X[/ilmath] is a collection of subsets, [ilmath]J\subseteq\mathcal{P}(X)[/ilmath][Note 1] such that[1][2]:

  • [ilmath]X\in\mathcal{J} [/ilmath] and [ilmath]\emptyset\in J[/ilmath]
  • If [ilmath]\{U_i\}_{i=1}^n\subseteq\mathcal{J}[/ilmath] is a finite collection of elements of [ilmath]\mathcal{J} [/ilmath] then [ilmath]\bigcap_{i=1}^nU_i\in\mathcal{J}[/ilmath] too - [ilmath]\mathcal{J} [/ilmath] is closed under finite intersection.
  • If [ilmath]\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J}[/ilmath] is any collection of elements of [ilmath]\mathcal{J} [/ilmath] (finite, countable, uncountable or otherwise) then [ilmath]\bigcup_{\alpha\in I}U_\alpha\in\mathcal{J}[/ilmath] - [ilmath]\mathcal{J} [/ilmath] is closed under arbitrary union.

We call the elements of [ilmath]\mathcal{J} [/ilmath] the open sets of the topology.

A topological space is simply a tuple consisting of a set (say [ilmath]X[/ilmath]) and a topology (say [ilmath]\mathcal{J} [/ilmath]) on that set - [ilmath](X,\mathcal{ J })[/ilmath].

Note: A topology may be defined in terms of closed sets - A closed set is a subset of [ilmath]X[/ilmath] whose complement is an open set. A subset of [ilmath]X[/ilmath] may be both closed and open, just one, or neither.

Terminology

  • For [ilmath]x\in X[/ilmath] we call [ilmath]x[/ilmath] a point (of the topological space [ilmath](X,\mathcal{ J })[/ilmath])[1]
  • For [ilmath]U\in\mathcal{J} [/ilmath] we call [ilmath]U[/ilmath] an open set (of the topological space [ilmath](X,\mathcal{ J })[/ilmath])[1]
(Unknown grade)
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
just find a glut and spew them here, the definition is the one thing every book I've found agrees on

Examples

Given a set [ilmath]X[/ilmath], the following topologies can be constructed:

If [ilmath](X,d)[/ilmath] is a metric space, then we have the:

If [ilmath](X,\preceq)[/ilmath] is a poset, then we have the:

See also

Notes

  1. Or [ilmath]\mathcal{J}\in\mathcal{P}(\mathcal{P}(X))[/ilmath] if you prefer, here [ilmath]\mathcal{P}(X)[/ilmath] denotes the power-set of [ilmath]X[/ilmath]. This means that if [ilmath]U\in\mathcal{J} [/ilmath] then [ilmath]U\subseteq X[/ilmath]

References

  1. 1.0 1.1 1.2 Introduction to Topological Manifolds - John M. Lee
  2. Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha