[permalink] [id link]

* In the cocountable topology on R ( or any uncountable set for that matter ), no infinite set is compact.

from
Wikipedia

## Some Related Sentences

cocountable and topology

The

__cocountable____topology__**(**also called**the**" countable complement__topology__")**on****any****set**X consists of**the**empty**set**and all__cocountable__subsets of X**.**
The

__cocountable____topology__**or**countable complement__topology__**on****any****set**X consists of**the**empty**set**and all__cocountable__subsets of X,**that****is**all sets whose complement in X**is**countable**.**
Every

**set**X with**the**__cocountable____topology__**is**Lindelöf, since every nonempty open**set**omits only countably many points of X**.**
However, in

**the**__cocountable____topology__all convergent sequences are eventually constant, so limits are unique**.**
The

__cocountable____topology__**on**an**uncountable****set****is**hyperconnected, thus connected, locally connected and pseudocompact, but neither weakly countably**compact**nor countably metacompact, hence not**compact****.**

cocountable and set

" Almost all "

**is**sometimes used synonymously with " all but finitely many "**(**formally, a cofinite__set__)**or**" all but a countable__set__"**(**formally, a__cocountable____set__); see almost**.****In**mathematics, a

__cocountable__subset of a

__set__X

**is**a subset Y whose complement in X

**is**a countable

__set__

**.**

The

__set__of all subsets of X**that**are either countable**or**__cocountable__forms a σ-algebra, i**.**e., it**is**closed under**the**operations of countable unions, countable intersections, and complementation**.**

cocountable and for

While

**the**rational numbers are a countable subset of**the**reals,__for__example,**the**irrational numbers are a__cocountable__subset of**the**reals**.**

topology and on

The

__topology__was a bus: cables were daisy-chained from each connected machine to**the**next, up to**the**maximum of 32 permitted__on__**any**LocalTalk segment**.**
The real line

**R**with its usual__topology__**is**a locally**compact**Hausdorff space, hence we can define a Borel measure__on__it**.**
Equipped with

**the**__topology__of pointwise convergence__on__A**(**i**.**e.,**the**__topology__induced by**the**weak -*__topology__of A < sup >∗</ sup >),**the**character space, Δ**(**A**),****is**a Hausdorff**compact**space**.**
For instance,

**any**continuous function defined__on__a**compact**space into an ordered**set****(**with**the**order__topology__) such as**the**real line**is**bounded**.*******Consider

**the**

**set**of all functions from

**the**real number line to

**the**closed unit interval, and define a

__topology__

__on__so

**that**a sequence in converges towards if and only if converges towards

**for**all

**.**

**In**mathematics, differential

__topology__

**is**

**the**field dealing with differentiable functions

__on__differentiable manifolds

**.**

Differential

__topology__considers**the**properties and structures**that**require only a smooth structure__on__a manifold to be defined**.**
To put it succinctly, differential

__topology__studies structures__on__manifolds which, in a sense, have**no**interesting local structure**.**
Likewise,

**the**problem of computing a quantity__on__a manifold which**is**invariant under differentiable mappings**is**inherently global, since**any**local invariant will be trivial in**the**sense**that**it**is**already exhibited in**the**__topology__of**R**< sup > n </ sup >.
Differential

__topology__also deals with questions like these, which specifically pertain to**the**properties of differentiable mappings__on__**R**< sup > n </ sup >**(****for**example**the**tangent bundle, jet bundles,**the**Whitney extension theorem, and so forth ).
Differential

__topology__**is****the**study of**the****(**infinitesimal, local, and global ) properties of structures__on__manifolds having**no**non-trivial local moduli, whereas differential geometry**is****the**study of**the****(**infinitesimal, local, and global ) properties of structures__on__manifolds having non-trivial local moduli**.**
The Euclidean

__topology__turns out to be equivalent to**the**product__topology____on__**R**< sup > n </ sup > considered as a product of n copies of**the**real line**R****(**with its standard__topology__).
An important result

__on__**the**__topology__of**R**< sup > n </ sup >,**that****is**far from superficial,**is**Brouwer's invariance of domain**.**

topology and R

**(**Note

**that**

**the**sets are open in

**the**subspace

__topology__even though they are not open as subsets of

__R__.)

Any subset of

__R__< sup > n </ sup >**(**with its subspace__topology__)**that****is**homeomorphic to another open subset of__R__< sup > n </ sup >**is**itself open**.*******

**the**topological vector space of all functions from

**the**real line

__R__to itself, with

**the**

__topology__of pointwise convergence

**.**

This example, in slightly different guises,

**is**important in algebraic geometry,__topology__and projective geometry where it may be denoted variously by PG**(**2,__R__**),**RP < sup > 2 </ sup >,**or**P < sub > 2 </ sub >(__R__) among other notations**.*******An example of a separable space

**that**

**is**not second-countable

**is**

__R__< sub > llt </ sub >,

**the**

**set**of real numbers equipped with

**the**lower limit

__topology__

**.**

Stone starts with an arbitrary

**compact**Hausdorff space X and considers**the**algebra C**(**X,__R__) of real-valued continuous functions**on**X, with**the**__topology__of uniform convergence**.**
The weak

__topology__**is**characterized by**the**following condition: a net**(**x < sub > λ </ sub >) in X converges in**the**weak__topology__to**the**element x of X if and only if φ**(**x < sub > λ </ sub >) converges to φ**(**x ) in__R__**or**C**for**all φ in X*******.****In**other words, it

**is**

**the**coarsest

__topology__such

**that**

**the**maps T < sub > x </ sub > from X

*****to

**the**base field

__R__

**or**C remain continuous

**.**

**In**abstract algebra and algebraic geometry,

**the**spectrum of a commutative ring

__R__, denoted by Spec

**(**

__R__

**),**

**is**

**the**

**set**of all proper prime ideals of

__R__

**.**It

**is**commonly augmented with

**the**Zariski

__topology__and with a structure sheaf, turning it into a locally ringed space

**.**

topology and any

Over

__any__vector space without__topology__, we may also notate**the**vectors by kets and**the**linear functionals by bras**.**
These categories surely have some objects

**that**are " special " in a certain way, such as**the**empty**set****or****the**product of two topologies, yet in**the**definition of a category, objects are considered to be atomic, i**.**e., we do not know whether an object A**is**a**set**, a__topology__,**or**__any__other abstract concept – hence,**the**challenge**is**to define special objects without referring to**the**internal structure of those objects**.****In**mathematics, specifically general

__topology__and metric

__topology__, a

**compact**space

**is**a mathematical space in which

__any__

**infinite**collection of points sampled from

**the**space must — as a

**set**— be arbitrarily close to some point of

**the**space

**.**

Slightly more generally,

__any__space with a finite__topology__**(**only finitely many open sets )**is****compact**; this includes in particular**the**trivial__topology__**.*******The spectrum of

__any__commutative ring with

**the**Zariski

__topology__

**(**

**that**

**is**,

**the**

**set**of all prime ideals )

**is**

**compact**, but never Hausdorff

**(**except in trivial cases ).

**In**mathematics, more specifically algebraic

__topology__,

**the**fundamental group

**(**defined by Henri Poincaré in his article Analysis Situs, published in 1895 )

**is**a group associated to

__any__given pointed topological space

**that**provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other

**.**

Some examples of groups

**that**are not Lie groups**(**except in**the**trivial sense**that**__any__group can be viewed as a 0-dimensional Lie group, with**the**discrete__topology__**),**are:**In**essence, a sequence

**is**a function with domain

**the**natural numbers, and in

**the**context of

__topology__,

**the**range of this function

**is**usually

__any__topological space

**.**

Pointless

__topology__then studies lattices like these abstractly, without reference to__any__underlying**set**of points**.**
The property of separability does not in and of itself give

__any__limitations**on****the**cardinality of a topological space:__any__**set**endowed with**the**trivial__topology__**is**separable, as well as second countable, quasi-compact, and connected**.****In**Star

__topology__every node

**(**computer workstation

**or**

__any__other peripheral )

**is**connected to central node called hub

**or**switch

**.**

Algebraic

__topology__,**for**example, allows**for**a convenient proof**that**__any__subgroup of a free group**is**again a free group**.**
This abstraction allows

__any__resource to be moved to a different physical location in**the**address__topology__of**the**network, globally**or**locally in an intranet**.**
::# The physical distributed bus

__topology__**is**sometimes incorrectly referred to as a physical tree__topology__– however, although**the**physical distributed bus__topology__resembles**the**physical tree__topology__, it differs from**the**physical tree__topology__in**that**there**is****no**central node to which__any__other nodes are connected, since this hierarchical functionality**is**replaced by**the**common bus**.**0.154 seconds.