But the facts about our Advisory Board and its members' duties are only one of several sets of facts about the quest for advice, both reliable and imaginative, on which to base our selections of Fellows.

But humans can do something equally useful, in the case of certain enumerably infinite sets: They can give explicit instructions for determining the nth member of the set, for arbitrary finite n. Such instructions are to be given quite explicitly, in a form in which they could be followed by a computing machine, or by a human who is capable of carrying out only very elementary operations on symbols.

Another equivalent axiom only considers collections X that are essentially powersets of other sets:

Its domain is the powerset of A ( with the empty set removed ), and so makes sense for any set A, whereas with the definition used elsewhere in this article, the domain of a choice function on a collection of sets is that collection, and so only makes sense for sets of sets.

For example, after having established that the set X contains only non-empty sets, a mathematician might have said " let F ( s ) be one of the members of s for all s in X.

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

The symbol Σ denotes a σ-algebra of sets, and Ξ denotes just an algebra of sets ( for spaces only requiring finite additivity, such as the ba space ).

For example, in Virginia, the definition of the conduct that constitutes the crime of robbery exists only in the common law, and the robbery statute only sets the punishment.

With longer sets ( more than ~ 40 people ) this would require long enough sets that the caller will usually only run the dance all the way around on ( rare ) non equal-turn dances.

A more refined classification is often shown in colored presentations of the periodic table ; this system restricts the terms " metal " and " nonmetal " to only certain of the more broadly defined metals and nonmetals, adding additional terms for certain sets of the more broadly viewed metals and nonmetals.

This is only effective for small sets, however ; for larger sets, this would be time-consuming and error-prone.

Specifically, a topological space is compact if, whenever a collection of open sets covers the space, some subcollection consisting only of finitely many open sets also covers the space.

It was this notion of compactness that became the dominant one, because it was not only a stronger property, but it could be formulated in a more general setting with a minimum of additional technical machinery, as it relied only on the structure of the open sets in a space.

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

Two sets have the same cardinal number if and only if there is a bijection between them.

A metric space X is complete if and only if every decreasing sequence of non-empty closed subsets of X, with diameters tending to 0, has a non-empty intersection: if F < sub > n </ sub > is closed and non-empty, for every n, and diam ( F < sub > n </ sub >) → 0, then there is a point x ∈ X common to all sets F < sub > n </ sub >.

Enumerative definitions are only possible for finite sets and only practical for relatively small sets.

Cantor himself only mentioned the ternary construction in passing, as an example of a more general idea, that of a perfect set that is nowhere dense.

( Konig is now remembered as having only pointed out that some sets cannot be well-ordered, in disagreement with Cantor.

Before Cantor, there were only finite sets ( which are easy to understand ) and " the infinite " ( which was considered a topic for philosophical, rather than mathematical, discussion ).

Under this aspect, the inclusion of actual infinity into mathematics, which explicitly started with G. Cantor only towards the end of the last century, seems displeasing.

A subset of Baire space is given the classification,, or if and only if the corresponding subset of Cantor space has the same classification.

Cantor was the only subject who was told of the surprise in advance ; he was recovering from a heart attack and it was felt that the shock might harm him.

In this way, results apply not only to the set of real numbers or set of integers but to more general time scales such as a Cantor set.

A subset of Baire space is given the classification,, or if and only if the corresponding subset of Cantor space has the same classification.

By considering only the points of the tori that are not removed at some stage, an embedding results of the sphere with a Cantor set removed.

* Cantor Charles Romalis, first and only Cantor of Temple Beth Tikvah in Wayne, NJ ( 1965 – present )

We then translate this function to the interval 5 / 8 so that the resulting function, which we call f < sub > 1 </ sub >, is nonzero only on the middle interval of the complement of the Smith – Volterra – Cantor set.

Of these, only George White's Scandals ( 1939 ) and Banjo Eyes with Eddie Cantor ( 1941 ) had long runs.

* 1940: Eddie Cantor, one of only two balloons based on a living person or people, The Tin Man

They did not discover the commercial center of Cantor, which was still several miles upriver ( it was only discovered a couple of years later, by Diogo Gomes ).

Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if and only if there exists a bijective function between them.

Vocaloid had no previous rival technology to contend with at the time of its release, with the English version only having to face the later release of VirSyn's Cantor software during its original run.

" He was therefore fanatically loyal to the king ," Norman Cantor observes, " and was concerned only with the full exercise of royal power.

As with many other episodes in the series, " Thirty Minutes Over Tokyo " suggests that, in the end, the Simpsons are more attached to the local than to the global, and, as Cantor writes: " indeed the global is ultimately important in the series only insofar as it can be made local, that is, part of Springfield.

Cantor comes to the jail to bail them out, but finds there is not a Yussel Rabinovitch there, only a Jess Robin.

It was the only thing in his life for which he felt guilt.

His looting of the orderly room had taken only a minute or two and the vicinity was still clear of guerrillas.

It was pitiful to see the thin ranks of warriors, old and young, wheeling and twisting their ponies frantically from side to side only to be tumbled bleeding from their saddles by the relentless slam, slam of the cruelly efficient Hawkinses.

The fire had gone down, and the man was only a shadow against the trees.

There was only one place where Jake Carwood's description had gone badly awry: the peace and quiet.

It was the only thing about her that was the least bit hard to remember.

under the circumstances I was only too willing to confess all.

only the counter at one end was lighted by a long fluorescent tube suspended directly above it.

On a shelf in the office behind the counter was a small radio dialed permanently on a station which broadcast only vulgar commercials and cheap popular music.

Once, pressing him, I learned that his job was only part-time, in the afternoons when nothing went on in the hall.

Though only a relatively short walk separated it from my own part of town, its character was wholly foreign to me.

The river was only a few blocks away but an unbroken line of piers prevented me from seeing it.

Although it was dark as usual I could see that the hall had only recently contained a great many people.

This desire, I went on, growing voluble as my conviction was aroused, had mounted at such a rate recently that I now found its realization necessary not only to my physical but also to my spiritual wellbeing.

The only reason we brought you was to get Miller out.

The only thing which would have attracted attention was that two wore the uniform of prison guards, three the striped suits of convicts.

He had belonged to this land and, perhaps, had desecrated it -- and this was the only material symbol that remained of him.

There was only one place where the mountain might receive her -- that unnamed, unnameable pool harbored in its secret bosom.

Now, he could only play the last card in what was probably the world's coldest deck.

He was only vaguely aware of the sluicing rain.

He paused only long enough to ascertain that Jess's buckskin was still missing and that his own gray was all right, then climbed through a back window and dropped to the ground outside.

There was no lock on the door, only an iron hook which he unfastened.

Again he stood in the darkness listening, but there was only the scrape of a shod hoof on a plank floor.