Title: Theory / Unimportant / CS / Fundamental / Automata / Infinite / Perrin Pin+10 / 03 05 topology
Tags: fromMath mathToDef closed mathToSee hasSeenConjDef toBinaryFunction

Name: A Polish space
Definition: which admits a countable basis of open sets.
Page: 140
Chapter: III
Section: 2.4
Definition2: completly metrizable
Conjdef: A topology

Smaller0: \(X\) has a countable basis of open sets
Greater0: \(X\) contains a countable dense subset
Page: 140
Chapter: III
Section: 2.4
Variables: \((X,\mathcal T)\) a topology
Index: 6
Kind: Prop
Smaller to greater: iff
Assuming: \(X\) is metrizable

Name: The class of Polish spaces
Page: 140
Section: 2.4
Chapter: III
Index: 7
Kind: Prop
Under: product
Condition: countable
Under2: union?

Name: the class of Borel sets
Definition: containing open sets
Page: 141
Chapter: III
Section: 2.5
Variables: \(E\) a topological space and
\(C\subseteq\mathcal P(E)\)
Index: 8
Kind: Prop
Definition2: closed under countable unions
Conjdef: the smallest class of subset of \(E\)
Definition3: closed under countable intersections
Assuming: \(E\) is metrizable

Name: The dual class of \(\Gamma\)
Definition: \(\{X ^c\mid X\in \Gamma\}\)
Page: 142
Chapter: III
Variables: \(E\) a set
\(\Gamma\subseteq \mathcal P(E)\).
Context: Sets
Notation: \(\neg\Gamma\)

Definition: \(\{\bigcup_{k\ge 0}X_k\mid X_k\in \Gamma\}\).
Page: 142
Chapter: III
Variables: \(E\) a topological space and
\(\Gamma\subseteq \mathcal P(E)\).
Notation: \(\exists^\omega\Gamma\)

Definition: \(\{\bigcap_{k\ge 0}X_k\mid X_k\in \Gamma\}\).
Page: 142
Chapter: III
Variables: \(E\) a topological space
\(\Gamma\subseteq \mathcal P(E)\).
Notation: \(\forall^\omega\Gamma\)

Definition: The class of open sets.
Page: 141
Chapter: III
Variables: \(E\) a topological space
Notation: \(\Sigma_1\)

Definition: \(\neg\Sigma_n\).
Page: 142
Chapter: III
Section: 2.5
Variables: \(E\) a topological space and \(n\ge 1\)
Notation: \(\Pi_{n}\)

Definition: \(\Sigma_n\cap\Pi_{n}\).
Page: 142
Chapter: III
Section: 2.5
Variables: \(E\) a topological space
\(n\ge 1\)
Notation: \(\Delta_n\)

Definition: \(\exists^\omega\Pi_{n-1}\).
Page: 142
Chapter: III
Section: 2.5
Variables: \(E\) a topological space and \(n\ge 2\)
Notation: \(\Sigma_n\)

Name: The Borel classes
Page: 143
Section: 2.5
Chapter: III
Index: 10
Kind: Prop
Under: inverse of continuous map
Under2: boolean?

Name: \(\Pi_k\)
Page: 143
Section: 2.5
Chapter: III
Index: 11
Kind: Prop
Under: product
Condition: countable

Name: Completly metrizable topological space
Definition: A topological space which is metrizable by a distance \(d\) and complete for \(d\).
Page: 137
Chapter: III
Section: 2

Name: The set of open sets of a Polish space
Page: 140
Section: 2.4
Chapter: III
Index: 4
Kind: Prop
Under: countable intersection of open subsets

Name: A chain
Definition: An increasing sequence of elements of an ordered set
Page: 223
Chapter: V
Section: 4.1

Page: 144, 145
Name: The topology of \(A^\omega\)
Set: \(A^\omega\)
Chapter: III
Section: 3
Variables: \(A\) an alphabet
Kind: Prop
Base: \(wA^\omega\) for \(w\) a word
Opens: \(XA^\omega\) with \(X\subseteq A^*\)
\(XA^\omega\) with \(X\) is a prefix subset of \(A^*\).
Metric: inverse of the size of the longest common prefix
Index: 1, 2, 4
Polish: yes if \(A\) countable

Name: The class of Borel sets
Definition: containing the open sets
Page: 141
Chapter: III
Section: 2.5
Variables: \(E\) a topological space
Definition2: closed under countable Boolean operations
Conjdef: The smallest class of sets

Page: 145
Section: 3
Notation: \(d(u,v)\)
Variables: \(A\) an alphabet,
\(u,v\in A^\omega\)
Chapter: III
Definition: \(2^{\min\{n\mid u(n)\neq v(n)\}}\)
Kind: Prop
Index: 2, 3
Metric: Complete metric

Smaller0: the closure of \(X\)
Greater0: the set of words \(w\in A^\omega\) whose every prefix are a prefix of a word of \(X\)
Page: 146
Chapter: III
Section: 3
Variables: \(A\) an alphabet,
\(X\subseteq A^\omega\)
Index: 5
Kind: Prop
Smaller to greater: equals
Context: \(A^\omega\)

Name: A closed set of \(A^\omega\)
Definition: \(\overrightarrow P\) for some \(P\subseteq A^*\)
Page: 146
Chapter: III
Section: 3
Variables: \(A\) an alphabet,
\(X\subseteq A^\omega\)
Index: 1, 6 7
Kind: Ex, Lem, Prop
Definition2: \(\overrightarrow P\) for some \(P\subseteq A^*\) prefix-closed
Example: \(B^\omega\) for \(B\subseteq A\).
Definition3: Given some \(P\subseteq A^*\): the set of infinite words with no prefix belonging to \(P\)
Example2: \(B^+C^\omega\cup B^\omega\) for \(B,C\) two subsets of \(A\).
Definition4: Some set \(X\subseteq A^\omega\) such that \(X=\overrightarrow {Pref(X)}\)
Definition5: A set recognized by a deterministic Büchi automaton in which each state is final
Example3: \(XA^\omega\) for \(X\) a set of words of the same length.

Smaller0: \(E\) is metrizable
Greater0: included into \(\Sigma_{n+1}\cap\Delta_{n+1}\)
Page: 142
Chapter: III
Section: 2.5
Variables: \(E\) a topological space
\(n\ge 1\)
Index: 9
Kind: Prop
Prefix Greater: \(\Delta_n\) is