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.

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

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

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

The anti de Sitter space of signature ( p, q ) can then be isometrically embedded in the space with coordinates ( x < sub > 1 </ sub >, ..., x < sub > p </ sub >, t < sub > 1 </ sub >, ..., t < sub > q + 1 </ sub >) and the pseudometric

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

The pseudometric topology is the topology induced by the open balls

* On the set of all non-empty subsets of M, d < sub > H </ sub > yields an extended pseudometric.

Experience is not seen, as it is in classical rationalism, as presenting us initially with clear and distinct objects simply located in space and registering their character, movements, and changes on the tabula rasa of an uninvolved intellect.

It is notable that at this time he was writing with admiration of Cimabue's and Poussin's way of filling space.

Next I refer to our program in space exploration, which is often mistakenly supposed to be an integral part of defense research and development.

Third, the United States is pressing forward in the development of large rocket engines to place vehicles of many tons into space for exploration purposes.

Like ours, the economy of the space merchants must constantly expand in order to survive, and, like ours, it is based on the principle of `` ever increasing everybody's work and profits in the circle of consumption ''.

And this, of course, is exactly what Madison Avenue has been accused of doing albeit in a primitive way, with its `` hidden persuaders '' and what the space merchants accomplish with much greater sophistication and precision.

Thus the copywriter in the world of the space merchants is the person who in earlier ages might have been a lyric poet, the person `` capable of putting together words that stir and move and sing ''.

But there is hope, for Conservation Commissioner Bontempo has tagged the sanctuary as the kind of place the state hopes to include in its program to double its park space.

Four hundred million dollars of the increase is for the expanded space program, a responsibility similarly neglected by Mr. Eisenhower.

I would like to see you devote some space in an early issue to the news blackout concerning President Kennedy's activities, so far as Southern California is concerned.

Advice is given also on problems of plant location and plant space.

The outside 4-inch space is filled by mortaring blocks on edge.

The space between them is filled with pit-run gravel or earth.

In general, such apartments afford more protection than smaller buildings because their walls are thick and there is more space.

The problem for the city apartment dweller is primarily to plan the use of existing space.

The music is always allowed the living space needed to attain its full sonority ; ;

But if space and money are no problem and small children are not on hand every day, it is certainly more restful to have your pool and entertainment area removed from the immediate environs of the house.

Since there is a continual loss of micrometeoritic material in space because of the radiation effects, there must be a continual replenishment: otherwise, micrometeorites would have disappeared from interplanetary space.

With detectors sensitive to three mass intervals and based on a few counts, the second and third Russian space probes indicate that the flux of the smallest particles detected is less than that of larger ones.

Of course, if there is a dust blanket around the Earth, the fluxes in interplanetary space should be less than the figures given here.

If ( remember this is an assumption ) the minimal polynomial for T decomposes Af where Af are distinct elements of F, then we shall show that the space V is the direct sum of the null spaces of Af.

Conversely, suppose that **ya is in the null space of Af.

The authors set about answering this fundamental question through a detailed investigation of the patient's ability, tactually, ( 1 ) to perceive figure and ( 2 ) to locate objects in space, with his eyes closed ( or turned away from the object concerned ).

* The space enclosed by a set of struts on a biplane, see biplane # Overview

The set L ( V, W ; X ) of all bilinear maps is a linear subspace of the space ( viz.

The Hilbert space of a spin-0 point particle is spanned by a " position basis ", where the label r extends over the set of all points in space.

Since a maximal ideal in A is closed, is a Banach algebra that is a field, and it follows from the Gelfand-Mazur theorem that there is a bijection between the set of all maximal ideals of A and the set Δ ( A ) of all nonzero homomorphisms from A to C. The set Δ ( A ) is called the " structure space " or " character space " of A, and its members " characters.

The film Blade Runner ( 1982 ), adapted from Philip K. Dick's Do Androids Dream of Electric Sheep ?, is set in 2019 in a dystopian future in which manufactured beings called replicants are slaves used on space colonies and are legal prey on Earth to various bounty hunters who " retire " ( kill ) them.

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.

The Bolzano – Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded.

) Note that the same set of points would not have, as an accumulation point, any point of the open unit interval ; hence that space cannot be compact.

In general topological spaces, however, the different notions of compactness are not necessarily equivalent, and the most useful notion, introduced by Pavel Alexandrov and Pavel Urysohn in 1929, involves the existence of certain finite families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

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.

In general topological spaces, however, the different notions of compactness are not equivalent, and the most useful notion of compactness — originally called bicompactness — involves families of open sets that " cover " the space in the sense that each point of the space must lie in some set contained in the family.

* Any finite topological space, including the empty set, is compact.

In fact, every compact metric space is a continuous image of the Cantor set.