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• Borel sigma-algebra
  ...is [[Sigma-algebra generated by|generated by]] the [[Open set|open sets]] of the [[Metric space|metric space]] {{M|(\mathbb{R},\vert\cdot\vert)}}. We de ...ets, but in measure theory this seems to denote the sets of half-open-half-closed rectangles, and it is too common to ignore</ref>
  5 KB (854 words) - 09:25, 6 August 2015
• Topological space/Definition
  </noinclude>A ''topological space'' is a set <math>X</math> coupled with a "topology", {{M|\mathcal{J} }} on <math>X</ma * A topology, {{M|\mathcal{J} }} is a collection of subsets of {{M|X}}, <math>\mathcal{J}\subseteq\mathcal{P}(X)</math> with the following
  1 KB (227 words) - 18:09, 20 April 2016
• Notes:Measures
  ...nions from a few authors to see what is what. Bogachev for example (author of [[Books:Measure Theory - Volume 1 - V. I. Bogachev]]) doesn't require that ...Measure'' - [[countably additive set function]] defined on an [[algebra of sets]], [[Real valued function|Real valued]]
  5 KB (771 words) - 03:06, 21 March 2016
• Notes:Measure theory plan
  * Introduce [[ring of sets]] ...mu} }}) - Introduce a (positive) extended real valued [[countably additive set function]], {{M|\bar{\mu} }} on that ring. This will be a pre-measure and t
  4 KB (619 words) - 19:28, 24 May 2016
• Hereditary sigma-ring
  ...rties. Additionally the "use" section requires expansion. Comment on power-set and sigma-algebra special case. Find out about related term, {{sigma|ideal} A ''hereditary {{sigma|ring}}'', {{M|\mathcal{H} }}, is a system of sets that is both [[hereditary (measure theory)|hereditary]] and a [[sigma-ring|
  1 KB (220 words) - 21:25, 19 April 2016
• Site projects:Patrolling topology/Task list
  * [[Convergence of a sequence]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21 * [[Equivalence of Cauchy sequences]] - '''DONE''' - [[User:Alec|Alec]] ([[User talk:Alec|talk
  4 KB (404 words) - 21:36, 30 September 2016
• Topological separation axioms
  ...cal spaces]] in general. [[Cardinality]] arguments are "weak" in the sense of they don't really provide a useful constraint on the topology. Consider for ...{M|\mathcal{K} }} is some topology on {{M|[0,1]\subset\mathbb{R} }} (power-set / discrete topology being a good example)
  4 KB (569 words) - 00:08, 4 May 2016
• Regular topological space/Definition
  ...V)]}} - (here {{M|C(\mathcal{J})}} denotes the [[closed set|closed sets]] of the [[topology]] {{M|\mathcal{J} }})
  574 B (93 words) - 23:53, 3 May 2016
• Normal topological space/Definition
  ...{{M|C(\mathcal{J})}} denotes the collection of [[closed set|closed sets]] of the [[topology]], {{M|\mathcal{J} }})
  468 B (73 words) - 00:00, 4 May 2016
• Urysohn's lemma
  ...M|E}} and {{M|F}} be a pair of ''[[disjoint]]'' [[closed set|closed sets]] of {{M|X}}, then{{rITTGG}}: {{Todo|Get a picture - the idea of this theorem is brilliant, once you see it!}}
  583 B (101 words) - 00:21, 4 May 2016
• Class of sets closed under set-subtraction properties
  #REDIRECT [[Properties of classes of sets closed under set-subtraction]]
  72 B (9 words) - 11:25, 21 August 2016
• Notes:Connected space
  ...can be expressed as the [[union]] of two [[disjoint]] and non-empty [[open sets]]{{rITTMJML}}. ...that are both [[open set|open]] and [[closed set|closed]] are the [[empty set]], {{M|\emptyset}}, and {{M|X}} itself<ref name="ITTMJML"/>.
  1 KB (188 words) - 21:33, 30 September 2016
• Notes:Basis for a topology
  ...J})}} a basis for the space is a collection, {{M|\mathcal{B} }} of subsets of {{M|X}} such that: ...hcal{B}[B\in\mathcal{J}]}} - the elements of {{M|\mathcal{B} }} are [[open set|open]] in {{M|X}}.
  5 KB (966 words) - 16:56, 27 June 2016
• Notes:Generalising the limit
  ...s just hard to phrase! You're dragging through the space with a 'sequence' of some sort and seeing what is left as you sweep along it. I want to distil t ...lies to topological spaces (of which the power set is - so it has a use in sets), there was no point in attempting further abstraction as "converges in a g
  6 KB (1,118 words) - 11:34, 30 July 2016
• Semi-ring of sets
  ...is almost a measure. A [[ring of sets]] is closed under all the elementary set operations. We also adopt the convention of calling anything that can be extended to either a [[pre-measure]] (and thus
  3 KB (508 words) - 17:25, 18 August 2016
• The ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions
  ...'' [[ring of sets generated by|generated by]]<ref group="Note">At the time of writing the ring generated by page doesn't exist so I write this note to cl ...tly the set {{M|\mathcal{F} }} and all finite [[union|unions]] of elements of {{M|\mathcal{F} }}{{rMIAMRLS}}.
  7 KB (1,398 words) - 18:33, 19 August 2016
• Types of set algebras/Type table
  ...e, if you have come here from a search engine you want to go to [[types of set algebras]]''' ! [[Ring of sets|Ring]]<ref name="PTACC">Probability Theory - A comprehensive course - secon
  4 KB (573 words) - 20:00, 19 August 2016
• Doctrine:Measure theory terminology
  ...epted...) need to be created. For now however I just want to note splicing sets}} ...ay}{c} {\bf\text{on hereditary }\sigma\text{-ring} }\\ {\bf\text{(or power-set if }\sigma\text{-ring is a }\sigma\text{ algebra)} } \end{array} }\\
  2 KB (333 words) - 17:49, 27 February 2017
• Hereditary system generated by
  ...of an arbitrary family of a type of system is a system of that type in and of itself.}} ...ning {{M|S}} we cannot be sure that either one is a subset (proper or not) of the other! Instead we mean smallest in the following sense:
  2 KB (371 words) - 23:36, 23 August 2016
• A subset of a topological space is open if and only if it is a neighbourhood to all of its points
  ...to other pages, gosh for something so simple it's long! Also verification of proof}} ...xplains, some authors define a "neighbourhood to {{M|x}}" to be any [[open set]] containing {{M|x}}. We call this an "''open neighbourhood''". By default
  8 KB (1,529 words) - 00:27, 6 September 2016