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

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.

" 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.

While the rational numbers are a countable subset of the reals, for example, the irrational numbers are a cocountable subset of the reals.

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.

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.

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.

An example of a topological group which is not a Lie group is given by the rational numbers Q with the topology inherited from R. This is a countable space and it does not have the discrete topology.

An example is the set R with the topology generated by sets of the form U-C, where U is an open set in the usual sense, and C is any countable subset of U.

* Development ( topology ), a countable collection of open coverings

Ordinal-indexed sequences are more powerful than ordinary ( ω-indexed ) sequences to determine limits in topology: for example, ω < sub > 1 </ sub > ( omega-one, the set of all countable ordinal numbers, and the smallest uncountable ordinal number ), is a limit point of ω < sub > 1 </ sub >+ 1 ( because it is a limit ordinal ), and, indeed, it is the limit of the ω < sub > 1 </ sub >- indexed sequence which maps any ordinal less than ω < sub > 1 </ sub > to itself: however, it is not the limit of any ordinary ( ω-indexed ) sequence in ω < sub > 1 </ sub >, since any such limit is less than or equal to the union of its elements, which is a countable union of countable sets, hence itself countable.

In the mathematical field of topology, a G < sub > δ </ sub > set is a subset of a topological space that is a countable intersection of open sets.

In practical terms, this means that theorems of algebra and combinatorics are restricted to countable structures, while theorems of analysis and topology are restricted to separable spaces.

* second-countable space: the topology has a countable base

In fact, the countable complement topology is Lindelöf but neither σ-compact nor locally compact.

We cannot eliminate the Hausdorff condition ; a countable set with the indiscrete topology is compact, has more than one point, and satisfies the property that no one point sets are open, but is not uncountable.

An example of a countable double-pointed cofinite topology is the set of even and odd integers, with a topology that groups them together.

In the mathematical discipline of general topology, a Polish space is a separable completely metrizable topological space ; that is, a space homeomorphic to a complete metric space that has a countable dense subset.

The study of topology in mathematics extends all over through point set topology, algebraic topology, differential topology, and all the related paraphernalia, such as homology theory, homotopy theory.

* Base ( topology ) of a topology: a generating set of the open sets of the topology

* Cone ( topology ) of a set X, namely the union of all line segments connecting a fixed point to points of X

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.

But how can we define the empty set without referring to elements, or the product topology without referring to open sets?

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.

* R carrying the lower limit topology satisfies the property that no uncountable set is compact.

* 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 ).

* Let X be a simply ordered set endowed with the order topology.

* A is a topologically closed set in the norm topology of operators.

A set is open in the Euclidean topology if and only if it contains an open ball around each of its points.

Felix Hausdorff ( November 8, 1868 – January 26, 1942 ) was a German mathematician who is considered to be one of the founders of modern topology and who contributed significantly to set theory, descriptive set theory, measure theory, function theory, and functional analysis.