Difference between revisions of "Disconnected (topology)"
From Maths
(Created page with "{{Stub page|grade=C|msg=There's not much more to be said, but it does meet the defining criteria for a stub page, hence this}} : '''Note: ''' much more information may be foun...") |
(Added definition for a disconnected subset, equivalent conditions section to be transcluded from connected (topology) page) |
||
| Line 4: | Line 4: | ||
==[[/Definition|Definition]]== | ==[[/Definition|Definition]]== | ||
{{/Definition}} | {{/Definition}} | ||
| + | ===Disconnected subset=== | ||
| + | Let {{M|A\in\mathcal{P}(X)}} be an arbitrary [[subset of]] {{M|X}} (for a [[topological space]] {{Top.|X|J}} as given above), then we say {{M|A}} is ''disconnected in {{Top.|X|J}}'' if{{rITTBM}}: | ||
| + | * {{M|A}} is a [[disconnected topological space]] when considered with the [[subspace topology]] (from {{Top.|X|J}}) | ||
| + | ==[[Connected (topology)/Equivalent conditions|Equivalent conditions]]== | ||
| + | {{:Connected (topology)/Equivalent conditions}} | ||
==See also== | ==See also== | ||
* {{link|Connected|topology}} - a space is connected if it is not disconnected | * {{link|Connected|topology}} - a space is connected if it is not disconnected | ||
Latest revision as of 23:59, 30 September 2016
Stub grade: C
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
There's not much more to be said, but it does meet the defining criteria for a stub page, hence this
- Note: much more information may be found on the connected page, this page exists just to document disconnectedness.
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].
Disconnected subset
Let [ilmath]A\in\mathcal{P}(X)[/ilmath] be an arbitrary subset of [ilmath]X[/ilmath] (for a topological space [ilmath](X,\mathcal{ J })[/ilmath] as given above), then we say [ilmath]A[/ilmath] is disconnected in [ilmath](X,\mathcal{ J })[/ilmath] if[2]:
- [ilmath]A[/ilmath] is a disconnected topological space when considered with the subspace topology (from [ilmath](X,\mathcal{ J })[/ilmath])
Equivalent conditions
To a topological space [ilmath](X,\mathcal{ J })[/ilmath] being connected:
- A topological space is connected if and only if the only sets that are both open and closed in the space are the entire space itself and the emptyset
- Some authors give this as the definition of a connected space, eg[2]
- A topological space is disconnected if and only if there exists a non-constant continuous function from the space to the discrete space on two elements
- A topological space is disconnected if and only if it is homeomorphic to a disjoint union of two or more non-empty topological spaces
To an arbitrary subset, [ilmath]A\in\mathcal{P}(X)[/ilmath], being connected:
- Obviously, the only sets being both relatively open and relatively closed in [ilmath]A[/ilmath] are [ilmath]\emptyset[/ilmath] and [ilmath]A[/ilmath] itself. (This comes directly from the subspace definition above)
- A subset of a topological space is disconnected if and only if it can be covered by two non-empty-in-the-subset and disjoint-in-the-subset sets that are open in the space itself
- Then apply the definition above, a subset is considered connected if it is not disconnected
- A subset of a topological space is connected if and only if and only if the only two subsets that are both relatively open and relatively closed with respect to the subset are the empty-set and the subset itself
See also
- Connected - a space is connected if it is not disconnected
- Much more information is available on that page, this is simply a supporting page
References
- ↑ 1.0 1.1 Introduction to Topological Manifolds - John M. Lee
- ↑ 2.0 2.1 Introduction to Topology - Bert Mendelson
| |||||||||||||||||||||||
