Difference between revisions of "Topology"

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Should be easy to flesh out, find some more references and demote to grade C once acceptable

Definition

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

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

We call the elements of $\mathcal{J}$ the open sets of the topology.

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

Note: A topology may be defined in terms of closed sets - A closed set is a subset of

$X$ whose complement is an open set. A subset of

$X$ may be both closed and open, just one, or neither.

Terminology

For $x\in X$ we call $x$ a point (of the topological space $(X,\mathcal{ J })$ )^[1]
For $U\in\mathcal{J}$ we call $U$ an open set (of the topological space $(X,\mathcal{ J })$ )^[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

Notes

Jump up ↑ Or $\mathcal{J}\in\mathcal{P}(\mathcal{P}(X))$ if you prefer, here $\mathcal{P}(X)$ denotes the power-set of $X$ . This means that if $U\in\mathcal{J}$ then $U\subseteq X$

References

↑ ^{Jump up to: 1.0} ^1.1 ^1.2 Introduction to Topological Manifolds - John M. Lee

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

[ITTMJML-2] {Jump up to: 1.0} ^1.1 ^1.2 Introduction to Topological Manifolds - John M. Lee

[Note 1]

[1]

@@ Line 1: / Line 1: @@
-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.
+{{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.
-==Comparing topologies==
+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}}
+: '''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===
+==Terminology==
-Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/>
+* 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>
+* 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}}
-'''Smaller''' is a good way to remember this as there are 'less things' in the smaller topology.
+==Notes==
+<references group="Note"/>
-===Finer, Larger, Stronger===
+==References==
-Given two topologies <math>\mathcal{J}</math>, <math>\mathcal{K}</math> on {{M|X}} we say:<br/>
+<references/>
-<math>\mathcal{J}</math> is '''finer, larger''' or '''stronger''' than <math>\mathcal{K}</math> if <math>\mathcal{J}\supset\mathcal{K}</math>
+{{Topology navbox|plain}}
+{{Definition|Topology|Metric Space}}
-'''Larger''' is a good way to remember this as there are 'more things' in the larger topology.
-==Building new topologies==
-There are a few common ways to make new topologies from old:
-# [[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
-# [[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} }}
-# [[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}}
-==Common topologies==
-===Discreet topology===
-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}}.
-That is ''every'' subset of {{M|X}} is an open set of the topology
-===Indiscreet Topology===
-Given a set {{M|X}} the indiscreet topology on {{M|X}} is the topology {{M|(X,\{\emptyset,X\})}}
-[[Category:Topology]]
-[[Category:Subjects]]

Difference between revisions of "Topology"

Revision as of 20:49, 12 May 2016

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