Difference between revisions of "Topology"
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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}} | |
| + | __TOC__ | ||
| + | {{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]] {{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}}: | ||
| + | * {{M|X\in\mathcal{J} }} and {{M|\emptyset\in J}} | ||
| + | * 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]]. | ||
| + | * 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]]. | ||
| + | We call the elements of {{M|\mathcal{J} }} the [[open set|open sets]] of the topology. | ||
| − | + | 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}}. | |
| − | + | : '''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. | |
| − | + | ==Terminology== | |
| − | + | * For {{M|x\in X}} we call {{M|x}} a ''point'' (of the topological space {{Top.|X|J}})<ref name="ITTMJML"/> | |
| − | + | * For {{M|U\in\mathcal{J} }} we call {{M|U}} an ''[[open set]]'' (of the topological space {{Top.|X|J}})<ref name="ITTMJML"/> | |
| − | + | {{Requires references|just find a glut and spew them here, the definition is the one thing every book I've found agrees on}} | |
| − | + | ==Notes== | |
| − | + | <references group="Note"/> | |
| − | + | ==References== | |
| − | + | <references/> | |
| − | + | {{Topology navbox|plain}} | |
| − | + | {{Definition|Topology|Metric Space}} | |
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Revision as of 20:49, 12 May 2016
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
Contents
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]:
- [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:
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
Notes
- ↑ 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
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