Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied – that is to say every room contains a guest.

For example, every subset of Cantor or Baire space is a set ( that is, a set which equals the intersection of countably many open sets ).

* The interior of every union of countably many closed nowhere dense sets is empty.

* Every non-empty Baire space is of second category in itself, and every intersection of countably many dense open subsets of X is non-empty, but the converse of neither of these is true, as is shown by the topological disjoint sum of the rationals and the unit interval 1.

It also suffices to prove that every infinite set is Dedekind-infinite ( equivalently: has a countably infinite subset ).

Every set X with the cocountable topology is Lindelöf, since every nonempty open set omits only countably many points of X.

exists for every set X in Σ, then μ is also countably additive.

* There exists a countable subset ( finite or countably infinite ) P of G, such that every element a ≠ 1 in G has a unique factorisation of the form

Further, for a countably infinite group G, it is possible to imagine an infinite array in which every row and every column corresponds to some element g of G, and where the element a * b is in the row corresponding to a and the column responding to b. In this situation too, the Latin Square property says that each row and each column of the infinite array will contain every possible value precisely once.

Many theorems which are provable using choice are of an elegant general character: every ideal in a ring is contained in a maximal ideal, every vector space has a basis, and every product of compact spaces is compact.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

Some authors require in addition that μ ( C ) < ∞ for every compact set C. If a Borel measure μ is both inner regular and outer regular, it is called a regular Borel measure.

Note that a locally finite Borel measure automatically satisfies μ ( C ) < ∞ for every compact set C.

* For every natural number, the-sphere is compact.

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

By the same construction, every locally compact Hausdorff space X is a dense subspace of a compact Hausdorff space having at most one point more than X ..

Note that, in a metric space, every compact subset is closed and bounded.

* If the metric space X is compact and an open cover of X is given, then there exists a number such that every subset of X of diameter < δ is contained in some member of the cover.

* A metric space ( or more generally any first-countable uniform space ) is compact if and only if every sequence in the space has a convergent subsequence.

) In particular, every continuous bijective map from a compact space to a Hausdorff space is a homeomorphism.

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

Since every compact Hausdorff space is a Tychonoff space, and every subspace of a Tychonoff space is Tychonoff, we conclude that any space possessing a Hausdorff compactification must be a Tychonoff space.

Abraham Robinson and Allen Bernstein proved that every polynomially compact linear operator on a Hilbert space has an invariant subspace.

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.

** The theorem that every Hilbert space has an orthonormal basis.

** On every infinite-dimensional topological vector space there is a discontinuous linear map.

This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).

They allowed weaving of beads by raising every other thread and inserting strung beads in the shed, the space between the lowered and raised threads.

The converse is not always true ; not every Banach space is a Hilbert space.

By a theorem of Gelfand and Naimark, given a B * algebra A there exists a Hilbert space H and an isometric *- homomorphism from A into the algebra B ( H ) of all bounded linear operators on H. Thus every B * algebra is isometrically *- isomorphic to a C *- algebra.

# X has a sub-base such that every cover of the space by members of the sub-base has a finite subcover ( Alexander's sub-base theorem )

A Hausdorff space is H-closed if every open cover has a finite subfamily whose union is dense.

It is possible, although highly doubtful, that he killed none at all but merely let his reputation work for him by privately claiming every unsolved murder in the state.

Yet often fear persists because, even with the most rigid ritual, one is never quite free from the uneasy feeling that one might make some mistake or that in every previous execution one had been unaware of the really decisive act.

In the work of every artist, I suppose, there may be found one or more moments which strike the student as absolutely decisive, ultimately emblematic of what it is all about ; ;

he is questioning, also, every epistemology which stems from Hume's presupposition that experience is merely sense data in abstraction from causal efficacy, and that causal efficacy is something intellectually imputed to the world, not directly perceived.

As shown in Figure 1, there is a connection for communication between every pair of points.

Hence, the only defensible procedure is to repress any and every notion, unless it gives evidence that it is perfectly safe.

usually, this is most exasperating to men, who expect every woman to verify their preconceived notions concerning her sex, and when she does not, immediately condemn her as eccentric and unwomanly.

This is a problem to be solved not by America alone, but also by every nation cherishing the same ideals and in position to provide help.

The steady purpose of our society is to assure justice, before God, for every individual.

`` History has this in common with every other science: that the historian is not allowed to claim any single piece of knowledge, except where he can justify his claim by exhibiting to himself in the first place, and secondly to any one else who is both able and willing to follow his demonstration, the grounds upon which it is based.

There is every reason to recognize that in the very last years of his life, as we shall see, Thompson did take the drug in carefully rationed doses to ease the pains of his illness, but the exact date at which this began has never been determined.

The religious quest is often intense and deep, and there are students on every campus who are seriously wrestling with the most profound questions of meaning and value.

His neighbors celebrated his return, even if it was only temporary, and Morgan was especially gratified by the quaint expression of an elderly friend, Isaac Lane, who told him, `` A man that has so often left all that is dear to him, as thou hast, to serve thy country, must create a sympathetic feeling in every patriotic heart ''.

Had U.S. warships not appeared off the Dominican coast, there is every possibility that the country would now be wracked by civil war.

In accordance with legislation passed at the last session of Congress, each Representative is authorized to deliver to the Post Office in bulk newsletters, speeches and other literature to be dropped in every letter box in his district.

Yet every Sunday we sinners go to that emergency room to receive first aid, and we leave unmindful that the man who ministered to us is a human being who suffers, too.

Generally, throughout the South, there is a growing impatience with the pattern of violence with which every step of desegregation is met.

The brush moves up and down and is small enough to clean every dental surface, including the back of the teeth.

Every legislator from Brasstown Bald to Folkston is going to have his every vote subjected to the closest scrutiny as a test of his political allegiances, not his convictions.