User:Leigh.Samphier/Topology/Completed
Summation
Definition:Summation over Finite Subset
Summation over Finite Subset is Well-Defined
Definition:Summation over Finite Index
Summation over Finite Index is Well-Defined
Summation over Union of Disjoint Finite Index Sets
Generalized Sum
Finite Generalized Sum Converges to Summation
Generalized Sum Restricted to Non-zero Summands
Corollary of Generalized Sum Restricted to Non-zero Summands
Generalized Sum with Finite Non-zero Summands
Absolute Net Convergence Equivalent to Absolute Convergence
Generalized Sum of Constant Zero Converges to Zero
Generalized Sum with Countable Non-zero Summands
Corollary of Generalized Sum with Countable Non-zero Summands
Bounded Generalized Sum is Absolutely Convergent
Absolutely Convergent Generalized Sum Converges to Supremum
Generalized Sum over Subset of Absolutely Convergent Generalized Sum is Absolutely Convergent
Inequality Rule for Real Convergent Nets
Inequality Rule for Absolutely Convergent Generalized Sums
Sum Rule for Convergent Generalized Sums
Generalized Sum over Union of Disjoint Index Sets
Absolutely Convergent Generalized Sum over Union of Disjoint Index Sets
Generalized Hilbert Sequence Space
Definition:Generalized Hilbert Sequence Space
Characterization of Generalized Hilbert Sequence Space
Generalized Hilbert Sequence Space is Metric Space
Nagata-Smirnov Metrization Theorem
Nagata-Smirnov Metrization Theorem
Nagata-Smirnov Metrization Theorem/Necessary Condition
Nagata-Smirnov Metrization Theorem/Sufficient Condition
Nagata-Smirnov Metrization Theorem/Lemma 1
Nagata-Smirnov Metrization Theorem/Lemma 2
Rings of Continuous Functions
Combination Theorem for Bounded Real-Valued Functions
Combination Theorem for Continuous Real-Valued Functions
Combination Theorem for Bounded Continuous Real-Valued Functions
Definition:Ring of Continuous Mappings
Ring of Continuous Mappings is Subring of All Mappings
Zero of Ring of Continuous Mappings
Additive Inverse in Ring of Continuous Mappings
Unity of Ring of Continuous Mappings
Commutativity of Ring of Continuous Mappings
Definition:Ring of Continuous Real-Valued Functions
Ring of Continuous Real-Valued Functions is Ring
Additive Inverse in Ring of Continuous Real-Valued Functions
Zero of Ring of Continuous Real-Valued Functions
Unity of Ring of Continuous Real-Valued Functions
Ring of Continuous Real-Valued Functions is Commutative
Definition:Ring of Bounded Continuous Real-Valued Functions
Ring of Bounded Continuous Functions is Subring of Continuous Real-Valued Functions
Zero of Ring of Bounded Continuous Real-Valued Functions
Additive Inverse in Ring of Bounded Continuous Real-Valued Functions
Unity of Ring of Bounded Continuous Real-Valued Functions
Ring of Bounded Continuous Real-Valued Functions is Commutative
Ring of Bounded Continuous Functions is Ring of Continuous Functions for Pseudocompact Space
Ring of Bounded Continuous Functions is Ring of Continuous Functions for Compact Space
Definition:Lattice of Real-Valued Functions
Lattice of Real-Valued Functions forms Distributive Lattice
Lattice and Ring of Real-Valued Functions forms Ordered Ring
Definition:Lattice of Continuous Real-Valued Functions
Lattice of Continuous Functions is Sublattice of All Real-Valued Functions
Definition:Lattice of Bounded Continuous Real-Valued Functions
Lattice of Bounded Continuous Functions is Sublattice of Continuous Real-Valued Functions
Sober Spaces
Equal Relative Complements iff Equal Subsets
Relative Complement inverts Subsets of Relative Complement
Definition:Meet-Irreducible Open Set
Complement of Singleton Closure is Meet-Irreducible
Definition:System of Open Neighborhoods
System of Open Neighborhoods is a Completely Prime Filter
Definition:Sober Space/Definition 1
Definition:Sober Space/Definition 2
Definition:Sober Space/Separation vs Completeness
Meet-Irreducible Open Set Induces Completely Prime Filter
Completely Prime Filter Induces Meet-Irreducible Open Set
Complement of Closed Set is Open Set
Open Set Not in System of Open Neighborhoods Iff Subset of Complement of Singleton Closure
Union of Open Sets Not in System of Open Neighborhoods is Complement of Singleton Closure
System of Open Neighborhoods are Equal Iff Singleton Closures are Equal
Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods
Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Necessary Condition
Sober Space iff Completely Prime Filter is Unique System of Open Neighborhoods/Sufficient Condition
Definition:Generic Point of Topological Space
Definition:Generic Point of Topological Space/Definition 1
Definition:Generic Point of Topological Space/Definition 2
Equivalence of Definitions of Generic Point of Topological Space
Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace
Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace/Necessary Condition
Meet-Irreducible Open Set iff Complement is Closed Irreducible Subspace/Sufficient Condition
Equivalence of Definitions of Sober Space
Frames and Locales
Topology with Set Inclusion Forms a Frame
Topology with Set Inclusion Forms a Frame/Corollary
Topology with Set Inclusion Forms a Locale
Other
Restriction of Restriction is Restriction
Countably Infinite Set has Enumeration
Characterization of T1 Space using Neighborhood Basis
Characterization of T1 Space using Basis
Definition:Pointwise Negation of Real-Valued Function
Definition:Pointwise Difference of Real-Valued Functions
Pointwise Difference is Pointwise Addition with Negation
Absolute Value of Function is Composite with Absolute Value Function
Characterization of Pointwise Maximum of Real-Valued Functions
Characterization of Pointwise Minimum of Real-Valued Functions
Constant Real-Valued Function is Bounded
Max Operation Preserves Total Ordering
Min Operation Preserves Total Ordering
Real Numbers form Valued Field
Absolute Value Function is Continuous
Subring Containing Ring Unity has Unity
Subring of Commutative Ring is Commutative
Compact Space is Pseudocompact
Continuous Real-Valued Function on Compact Space is Bounded
Structure Induced by Idempotent Operation is Idempotent
Structure Induced by Absorbing Operations is Absorbing
Structure Induced by Left Distributive Operation is Left Distributive
Structure Induced by Right Distributive Operation is Right Distributive
Structure Induced by Distributive Operation is Distributive
Structure Induced by Semilattice Operation is Semilattice