User:Leigh.Samphier/Topology
Topology (Completed)
- 1982: Peter T. Johnstone: Stone Spaces: Chapter $\text I$: Preliminaries, Definition $1.2$ and $1.4$
- 2012: Jorge Picado and Aleš Pultr: Frames and Locales: Chapter $1$: Spaces and Lattices of Open Sets, $\S 1$ Sober spaces, Definition $1.1$
To Be Published
Sober Spaces
**User:Leigh.Samphier/Topology/Definition:Frame of Topological Space
**User:Leigh.Samphier/Topology/Frame of Topological Space is Frame rename Topology with Set Inclusion Forms a Frame
**User:Leigh.Samphier/Topology/Definition:Frame Homomorphism of Continuous Mapping
**User:Leigh.Samphier/Topology/Frame Homomorphism of Continuous Mapping is Frame Homomorphism
**User:Leigh.Samphier/CategoryTheory/Definition:Frame of Open Sets Functor
User:Leigh.Samphier/CategoryTheory/Preimage of Subset under Identity Mapping
User:Leigh.Samphier/CategoryTheory/Frame of Open Sets Functor is Contravariant
**User:Leigh.Samphier/Topology/Frame Homomorphism is Lower Adjoint of Unique Galois Connection
**User:Leigh.Samphier/Topology/Definition:Locale (Lattice Theory)
**User:Leigh.Samphier/Topology/Definition:Locale (Lattice Theory)/Frames vs Locales
**User:Leigh.Samphier/Topology/Definition:Continuous Map (Locale)
**User:Leigh.Samphier/Topology/Definition:Continuous Map (Locale)/Localic Mapping
**User:Leigh.Samphier/Topology/Definition:Localic Mapping
**User:Leigh.Samphier/Topology/Definition:Continuous Map (Locale)/Also Defined As
**User:Leigh.Samphier/Topology/Definition:Category of Locales
**User:Leigh.Samphier/Topology/Definition:Category of Locales with Localic Mappings
User:Leigh.Samphier/Topology/Upper Adjoint of Frame Homomorphism is Localic Mapping
User:Leigh.Samphier/Topology/Composite Localic Mapping is Localic Mapping
**User:Leigh.Samphier/Topology/Identity Mapping is Localic Mapping
**User:Leigh.Samphier/Topology/Category of Locales with Localic Mappings is Category
User:Leigh.Samphier/Topology/Definition:Locale of Topological Space
User:Leigh.Samphier/Topology/Locale of Topological Space is Locale rename Topology with Set Inclusion Forms a Locale
User:Leigh.Samphier/Topology/Continuous Mapping Induces Continuous Map of Locales
User:Leigh.Samphier/Topology/Definition:Continuous Map of Locales Induced by Continuous Mapping
User:Leigh.Samphier/CategoryTheory/Definition:Locale of Open Sets Functor
User:Leigh.Samphier/CategoryTheory/Locale of Open Sets Functor is Functor
User:Leigh.Samphier/CategoryTheory/Definition:Locallic Functor
User:Leigh.Samphier/CategoryTheory/Localic Functor is Functor
User:Leigh.Samphier/Topology/Definition:Spectrum of Locale
User:Leigh.Samphier/Topology/Spectrum of Locale is Sober Space
User:Leigh.Samphier/CategoryTheory/Definition:Spectrum Functor
User:Leigh.Samphier/CategoryTheory/Spectrum Functor is Functor
User:Leigh.Samphier/Topology/Spectrum of Locale of Sober Space is Homeomorphic
User:Leigh.Samphier/Topology/Locale of Spectrum of Spatial Locale is Isomorphic
$T_D$ Spaces
$T_D$ Spaces and Sober Spaces
User:Leigh.Samphier/Topology/TD Space need not be Sober
User:Leigh.Samphier/Topology/Sober Space need not be TD
User:Leigh.Samphier/Topology/T1 Space need not be Sober
User:Leigh.Samphier/Topology/Sober Space need not be T1
User:Leigh.Samphier/Topology/T1 and Sober Space need not be T2
Stone-Čech Compactification
User:Leigh.Samphier/Topology/Definition:Category of Compact Hausdorff Spaces
uniqueness-of-left-adjoint [[1]]
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Locales Johnstone
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Topological Spaces Johnstone
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Tychonoff Spaces Munkres
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Product of Real Unit Intervals Kelly, Munkres
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Product of Real Intervals Willard
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Family of z-Ultrafilters Gilman and Jerrison
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Largest Compactification Engelking
User:Leigh.Samphier/Topology/Definition:Stone-Čech Compactification/Maximal Ideal Space Walker
Fully T4
User:Leigh.Samphier/Topology/Definition:Star (Topology)
User:Leigh.Samphier/Topology/Definition:Star Refinement
User:Leigh.Samphier/Topology/Definition:Barycentric Refinement
User:Leigh.Samphier/Topology/Fully T4 Space is T4 Space
User:Leigh.Samphier/Topology/Fully Normal Space is Paracompact
User:Leigh.Samphier/Topology/Discrete Space is Fully T4
User:Leigh.Samphier/Topology/Metric Space is Fully T4
User:Leigh.Samphier/Topology/T3 Lindelöf Space is Fully T4 Space
User:Leigh.Samphier/Topology/Characterization of Paracompactness in T3 Space
User:Leigh.Samphier/Topology/T3 Space is Fully T4 iff Paracompact
User:Leigh.Samphier/Topology/T1 Space is Fully T4 iff Paracompact
Metrization
Stone-Weierstrass Theorem
- Stone-Weierstrass Theorem - Willard: General Topology
User:Leigh.Samphier/Topology/Definition:Algebra of Real-Valued Functions
User:Leigh.Samphier/Topology/Definition:Algebra of Continuous Real-Valued Functions
User:Leigh.Samphier/Topology/Definition:Algebra of Bounded Continuous Real-Valued Functions
User:Leigh.Samphier/Topology/Definition:Banach Algebra of Bounded Continuous Real-Valued Functions
Other
- Stone's Representation Theorem for Boolean Algebras - M H Stone: Various papers
- Jordan Curve Theorem - Various PDF
- Generalized Hilbert Sequence Space is Hilbert Space - Conway: A Course in Functional Analysis
- Hilbert Sequence Space is Hilbert Space - Conway: A Course in Functional Analysis