Disconnected (topology)/Definition

From Maths
Jump to: navigation, search
Grade: A
This page requires references, it is on a to-do list for being expanded with them.
Please note that this does not mean the content is unreliable, it just means that the author of the page doesn't have a book to hand, or remember the book to find it, which would have been a suitable reference.
The message provided is:
Mendelson and Lee's topological manifolds have it covered, I think Munkres is where I got "separation" from

Definition

A topological space, [ilmath](X,\mathcal{ J })[/ilmath], is said to be disconnected if[1]:

  • [ilmath]\exists U,V\in\mathcal{J}[U\ne\emptyset\wedge V\neq\emptyset\wedge V\cap U=\emptyset\wedge U\cup V=X][/ilmath], in words "if there exists a pair of disjoint and non-empty open sets, [ilmath]U[/ilmath] and [ilmath]V[/ilmath], such that their union is [ilmath]X[/ilmath]"

In this case, [ilmath]U[/ilmath] and [ilmath]V[/ilmath] are said to disconnect [ilmath]X[/ilmath][1] and are sometimes called a separation of [ilmath]X[/ilmath].

References

  1. 1.0 1.1 Introduction to Topological Manifolds - John M. Lee