) This is only a pseudometric, not yet a metric, since two different Cauchy sequences may have the distance 0.

This turns the seminormed space into a pseudometric space ( notice this is weaker than a metric ) and allows the definition of notions such as continuity and convergence.

* Every metric space is Tychonoff ; every pseudometric space is completely regular.

More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is a Baire space.

In mathematics, a pseudometric space is a generalized metric space in which the distance between two distinct points can be zero.

In the same way as every normed space is a metric space, every seminormed space is a pseudometric space.

A pseudometric space is a set together with a non-negative real-valued function ( called a pseudometric ) such that, for every,

Unlike a metric space, points in a pseudometric space need not be distinguishable ; that is, one may have for distinct values.

More generally, every topological space which is homeomorphic to an open subset of a complete pseudometric space is a Baire space.

Intuitively, this has the consequence that all points of the space are " lumped together " and cannot be distinguished by topological means ; it belongs to a pseudometric space in which the distance between any two points is zero.

This point then induces a pseudometric on the space of functions, given by

** In the product topology, the closure of a product of subsets is equal to the product of the closures.

** In topology, morphisms between topological spaces are called continuous maps, and an automorphism of a topological space is a homeomorphism of the space to itself, or self-homeomorphism ( see homeomorphism group ).

The spaceX * of all linear maps into K ( which is called the algebraic dual space to distinguish it from X ′) also induces a weak topology which is finer than that induced by the continuous dual since X ′ ⊆ X *.

* Corollary If X is a Banach space, then X is reflexive if and only if X ′ is reflexive, which is the case if and only if its unit ball is compact in 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.

The groupoid concept is important in topology.

Topos theory is a form of abstract sheaf theory, with geometric origins, and leads to ideas such as pointless topology.

A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps ( morphisms ) between topological spaces in topology ( the associated category is called Top ), and the study of smooth functions ( morphisms ) in manifold theory.

This is a fundamental idea, which first surfaced in algebraic topology.

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.

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.

A subset K of a topological space X is called compact if it is compact in the induced topology.

Slightly more generally, any space with a finite topology ( only finitely many open sets ) is compact ; this includes in particular the trivial topology.

Instead, with the topology of compact convergence, C ( a, b ) can be given the structure of a Fréchet space: a locally convex topological vector space whose topology can be induced by a complete translation-invariant metric.

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.

such that the topology induced by d is.

* A space X is completely regular if and only if it has the initial topology induced by C ( X ) or C *( X ).

Let ρ be the initial topology on X induced by C < sub > τ </ sub >( X ) or, equivalently, the topology generated by the basis of cozero sets in ( X, τ ).

If the field K has an absolute value, then the weak topology σ ( X, F ) is induced by the family of

The weak -* topology on X * is the weak topology induced by the image of T: T ( X ) ⊂ X **.

The Cauchy – Schwarz is used to prove that the inner product is a continuous function with respect to the topology induced by the inner product itself.

In fact, the collection of all left cosets ( or right cosets ) of C in G is equal to the collection of all components of G. Therefore, the quotient topology induced by the quotient map from G to G / C is totally disconnected.

If the topology of the topological vector space is induced by a metric which is homogenous, as in the case of a metric induced by the norm of normed vector spaces, then the two definitions coincide.

Given such an absolute value on K, a new induced topology can be defined on K. This topology is the same as the original topology.

One can show that the topology induced by the metric is the same as the product topology in the above definition.

Although constrained by land mass and topology, the amount of arable land, both regionally and globally, fluctuates due to human and climatic factors such as irrigation, deforestation, desertification, terracing, landfill, and urban sprawl.

More advanced questions involve the topology of the curve and relations between the curves given by different equations.

RTMP was the protocol by which routers kept each other informed about the topology of the network.

PNNI uses the same shortest-path-first algorithm used by OSPF and IS-IS to route IP packets to share topology information between switches and select a route through a network.

Over any vector space without topology, we may also notate the vectors by kets and the linear functionals by bras.

Categories were first introduced by Samuel Eilenberg and Saunders Mac Lane in 1942 – 45, in connection with algebraic topology.

Not every countably compact space is compact ; an example is given by the first uncountable ordinal with the order topology.

Not every compact space is sequentially compact ; an example is given by 2 < sup ></ sup >, with the product topology.

Modern computer buses can use both parallel and bit serial connections, and can be wired in either a multidrop ( electrical parallel ) or daisy chain topology, or connected by switched hubs, as in the case of USB.

Differential topology and differential geometry are first characterized by their similarity.

In one view, differential topology distinguishes itself from differential geometry by studying primarily those problems which are inherently global.

The ambiguous character of the axioms as originally formulated by Euclid makes it possible for different commentators to disagree about some of their other implications for the structure of space, such as whether or not it is infinite ( see below ) and what its topology is.

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.

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure ( e. g. inner product, norm, topology, etc.

The emergence of the code is governed by the topology defined by the probable errors and is related to the map coloring problem.

In condensed matter physics, the three dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms.