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Page "Compact space" ¶ 30
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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 a countable set is the discrete topology.
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.
If the complement is not finite, but it is countable, then one says the set is cocountable.
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 continuous dual space can be used to define a new topology on X: the weak topology.
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.
* The right order topology or left order topology on any bounded totally ordered set is compact.
* 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 metric topology on E < sup > n </ sup > is called the Euclidean topology.
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
* R carrying the lower limit topology satisfies the property that no uncountable set is compact.
( 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.

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