Difference between revisions of "Topology"
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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}} | {{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== | ==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}}: | + | 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}}: |
* {{M|X\in\mathcal{J} }} and {{M|\emptyset\in J}} | * {{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_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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* For {{M|U\in\mathcal{J} }} we call {{M|U}} an ''[[open set]]'' (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}} | {{Requires references|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 {{M|X}}, the following topologies can be constructed: | ||
| + | * [[Discrete topology]] - the topology here is {{M|\mathcal{P}(X)}} - the [[power set]] of {{M|X}}. | ||
| + | * [[Indiscrete topology]] ({{AKA}}: [[Trivial topology]]) - the only open sets are {{M|X}} itself and {{M|\emptyset}} | ||
| + | * [[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} }} | ||
| + | If {{M|(X,d)}} is a [[metric space]], then we have the: | ||
| + | * [[Metric topology]] ({{AKA}}: [[topology induced by a metric]]) - whose open sets are exactly the ones we consider open in a metric sense | ||
| + | ** This uses [[open balls]] as a [[topological basis]] | ||
| + | If {{M|(X,\preceq)}} is a [[poset]], then we have the: | ||
| + | * [[Order topology]] | ||
| + | ==See also== | ||
| + | * [[Topological separation axioms]] | ||
| + | ** Covers things like [[Hausdorff space]], [[Normal topological space]], so forth. | ||
==Notes== | ==Notes== | ||
<references group="Note"/> | <references group="Note"/> | ||
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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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:
- Discrete topology - the topology here is [ilmath]\mathcal{P}(X)[/ilmath] - the power set of [ilmath]X[/ilmath].
- Indiscrete topology (AKA: Trivial topology) - the only open sets are [ilmath]X[/ilmath] itself and [ilmath]\emptyset[/ilmath]
- Finite complement topology - the open sets are [ilmath]\emptyset[/ilmath] and any set [ilmath]U\in\mathcal{P}(X)[/ilmath] such that [ilmath]\vert X-U\vert\in\mathbb{N} [/ilmath]
If [ilmath](X,d)[/ilmath] is a metric space, then we have the:
- Metric topology (AKA: topology induced by a metric) - whose open sets are exactly the ones we consider open in a metric sense
- This uses open balls as a topological basis
If [ilmath](X,\preceq)[/ilmath] is a poset, then we have the:
See also
- Topological separation axioms
- Covers things like Hausdorff space, Normal topological space, so forth.
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
- ↑ 1.0 1.1 1.2 Introduction to Topological Manifolds - John M. Lee
- ↑ Functional Analysis - Volume 1: A gentle introduction - Dzung Minh Ha
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