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Order the natural numbers as follows: Now let F be a continuous function from the reals to the reals and suppose p≺q in the above ordering. Then if F has a point of least ...

A graph H is a minor of a graph G if a copy of H can be obtained from G via repeated edge deletion and/or edge contraction. The Kuratowski reduction theorem states that any ...

The converse of Pascal's theorem, which states that if the three pairs of opposite sides of (an irregular) hexagon meet at three collinear points, then the six vertices lie ...

A chordless graph is a simple graph possessing no chords. A chordal graph (which possesses no chordless cycles) is not the same as (or converse of) a chordless graph (which ...

A topological space X such that for every closed subset C of X and every point x in X\C, there is a continuous function f:X->[0,1] such that f(x)=0 and f(C)={1}. This is the ...

The expression im kleinen is German and means "on a small scale." A topological space is connected im kleinen at a point x if every neighborhood U of x contains an open ...

The term "continuum" has (at least) two distinct technical meanings in mathematics. The first is a compact connected metric space (Kuratowski 1968; Lewis 1983, pp. 361-394; ...

Let X be a set and S a collection of subsets of X. A set function mu:S->[0,infty] is said to possess countable monotonicity provided that, whenever a set E in S is covered by ...

A graph G is said to be disconnected if it is not connected, i.e., if there exist two nodes in G such that no path in G has those nodes as endpoints. The numbers of ...

A module over a unit ring R is called divisible if, for all r in R which are not zero divisors, every element m of M can be "divided" by r, in the sense that there is an ...