Difference between revisions of "Topology"

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Once you have understood [[Metric space|metric spaces]] you can read [[Motivation for topology|motivation for topology]] and see why [[Topological space|topological spaces]] "make sense" and extend metric spaces.
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{{Stub page|grade=A*|msg=Should be easy to flesh out, find some more references and demote to grade C once acceptable}}
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__TOC__
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{{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}}
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==Definition==
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A ''topology'' on a [[set]] {{M|X}} is a collection of [[subset|subsets]], {{M|J\subseteq\mathcal{P}(X)}}<ref group="Note">Or {{M|\mathcal{J}\in\mathcal{P}(\mathcal{P}(X))}} if you prefer, here {{M|\mathcal{P}(X)}} denotes the [[power-set]] of {{M|X}}. This means that if {{M|U\in\mathcal{J} }} then {{M|U\subseteq X}}</ref> such that{{rITTMJML}}{{rFAVIDMH}}:
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* {{M|X\in\mathcal{J} }} and {{M|\emptyset\in J}}
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* If {{M|1=\{U_i\}_{i=1}^n\subseteq\mathcal{J} }} is a [[finite]] collection of elements of {{M|\mathcal{J} }} then {{M|1=\bigcap_{i=1}^nU_i\in\mathcal{J} }} too - {{M|\mathcal{J} }} is [[closed]] under ''[[finite]]'' [[intersection]].
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* If {{M|1=\{U_\alpha\}_{\alpha\in I}\subseteq\mathcal{J} }} is ''any'' collection of elements of {{M|\mathcal{J} }} (finite, [[countable]], [[uncountable]] or otherwise) then {{M|1=\bigcup_{\alpha\in I}U_\alpha\in\mathcal{J} }} - {{M|\mathcal{J} }} is closed under ''[[arbitrary]]'' [[union]].
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We call the elements of {{M|\mathcal{J} }} the [[open set|open sets]] of the topology.
  
==Comparing topologies==
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A [[topological space]] is simply a [[tuple]] consisting of a set (say {{M|X}}) and a topology (say {{M|\mathcal{J} }}) on that set - {{Top.|X|J}}.
Let {{M|(X,\mathcal{J})}} and {{M|(X,\mathcal{K})}} be two [[Topological space|topologies]] on {{M|X}}
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: '''Note: ''' A [[Topology defined in terms of closed sets|topology may be defined in terms of closed sets]] - A [[closed set]] is a subset of {{M|X}} whose [[complement]] is an [[open set]]. A subset of {{M|X}} may be both closed and open, just one, or neither.
===Coarser, Smaller, Weaker===
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==Terminology==
Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/>
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* For {{M|x\in X}} we call {{M|x}} a ''point'' (of the topological space {{Top.|X|J}})<ref name="ITTMJML"/>
<math>\mathcal{J}</math> is '''coarser, smaller''' or '''weaker''' than <math>\mathcal{K}</math> if <math>\mathcal{J}\subset\mathcal{K}</math>  
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* For {{M|U\in\mathcal{J} }} we call {{M|U}} an ''[[open set]]'' (of the topological space {{Top.|X|J}})<ref name="ITTMJML"/>
 
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{{Requires references|just find a glut and spew them here, the definition is the one thing every book I've found agrees on}}
'''Smaller''' is a good way to remember this as there are 'less things' in the smaller topology.
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==Examples==
 
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Given a set {{M|X}}, the following topologies can be constructed:
===Finer, Larger, Stronger===
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* [[Discrete topology]] - the topology here is {{M|\mathcal{P}(X)}} - the [[power set]] of {{M|X}}.
Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/>
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* [[Indiscrete topology]] ({{AKA}}: [[Trivial topology]]) - the only open sets are {{M|X}} itself and {{M|\emptyset}}
<math>\mathcal{J}</math> is '''finer, larger''' or '''stronger''' than <math>\mathcal{K}</math> if <math>\mathcal{J}\supset\mathcal{K}</math>  
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* [[Finite complement topology]] - the open sets are {{M|\emptyset}} and any set {{M|U\in\mathcal{P}(X)}} such that {{M|\vert X-U\vert\in\mathbb{N} }}
 
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If {{M|(X,d)}} is a [[metric space]], then we have the:
'''Larger''' is a good way to remember this as there are 'more things' in the larger topology.
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* [[Metric topology]] ({{AKA}}: [[topology induced by a metric]]) - whose open sets are exactly the ones we consider open in a metric sense
 
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** This uses [[open balls]] as a [[topological basis]]
==Building new topologies==
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If {{M|(X,\preceq)}} is a [[poset]], then we have the:
There are a few common ways to make new topologies from old:
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* [[Order topology]]
# [[Product topology|Product]] Given topological spaces {{M|(X,\mathcal{J})}} and {{M|(Y,\mathcal{K})}} there is a topology on {{M|X\times Y}} called "the product topology" (the coarsest topology such that the [[Projection map|projections]] are continuous
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==See also==
# [[Quotient topology|Quotient]] Given a topological space {{M|(X,\mathcal{J})}} and an [[Equivalence relation|equivalence relation]] {{M|\sim}} on {{M|X}}, we can define the quotient topology on {{M|X}} which we often denote by {{M|\frac{\mathcal{J} }{\sim} }}  
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* [[Topological separation axioms]]
# [[Subspace topology|Subspace]] Given a topological space {{M|(X,\mathcal{J})}} and any {{M|Y\subset X}} then the topology on {{M|X}} can induce the subspace topology on {{M|Y}}
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** Covers things like [[Hausdorff space]], [[Normal topological space]], so forth.
 
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==Notes==
==Common topologies==
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<references group="Note"/>
===Discreet topology===
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==References==
Given a set {{M|X}} the Discreet topology on {{M|X}} is {{M|\mathcal{P}(X)}}, that is {{M|(X,\mathcal{P}(X))}} is the discreet topology on {{M|X}} where {{M|\mathcal{P}(X)}} is the [[Power set|power set]] of {{M|X}}.
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<references/>
 
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{{Topology navbox|plain}}
That is ''every'' subset of {{M|X}} is an open set of the topology
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{{Definition|Topology|Metric Space}}
 
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===Indiscreet Topology===
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Given a set {{M|X}} the indiscreet topology on {{M|X}} is the topology {{M|(X,\{\emptyset,X\})}}
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[[Category:Topology]]
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[[Category:Subjects]]
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Latest revision as of 09:28, 30 December 2016

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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]
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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