Difference between revisions of "TOP (category)"

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Latest revision as of 20:10, 20 February 2016

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Definition

[ilmath]\mathrm{TOP} [/ilmath] is the category of all topological spaces, the objects are tuples of a set [ilmath]X[/ilmath] and a topology [ilmath]\mathcal{J}_X[/ilmath] on [ilmath]X[/ilmath] and the arrows, or morphisms of the category are continuous functions[1]. More explicitly.

  • The objects of [ilmath]\mathrm{TOP} [/ilmath] are all topological spaces, [ilmath](X,\mathcal{J}_X)[/ilmath]
  • The arrows/morphisms of [ilmath]\mathrm{TOP} [/ilmath] are the continuous functions between spaces.

Discussion

TODO: Discuss as a subcategory of [ilmath]\mathrm{SET} [/ilmath], remember it must first go under the forgetful functor to discard the topological structure and distill it to just sets and mappings

References

  1. ↑ An Introduction to Category Theory - Harold Simmons - 1st September 2010 edition

v • d • e
 
Category Theory
Overview of the concepts of Category Theory
Key objects
[ilmath]\xymatrix{ & \text{Arrow} \\ \text{Monic} \ar@{^{(}->}[ur] & & \text{Epic} \ar@{^{(}->}[ul] \\ & \text{Bimorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \\ {\begin{array}{c}\text{Section}\\ \text{(Split monic)} \end{array} } \ar@{^{(}->}[uu] & & {\begin{array}{c}\text{Retraction}\\ \text{(Split epic)} \end{array} } \ar@<-0.75ex>@{^{(}->}[uu] \\ & \text{Isomorphism} \ar@{^{(}->}[ur] \ar@<-0.5ex>@{^{(}->}[ul] \ar@{^{(}->}[uu] }[/ilmath]
Typical morphism types
(see diagram on right)
Key objects
Primitive constructs
Key constructs
Important examples
Trivial category examples
Common categories

v • d • e
Topology
Overview of the structures studied in topology
Primitives
Global properties of topological spaces
Local properties of topological spaces
Constructs
Subtypes of topological spaces
inner product spaces [ilmath]\subset[/ilmath] normed spaces [ilmath]\subset[/ilmath] metric spaces [ilmath]\subset[/ilmath] topological spaces
Maps between topological spaces


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