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Maps between CW-complexes that induce isomorphisms on all homotopy groups are actually homotopy equivalences.

A theorem that classifies planar regular closed curves up to regular homotopy by their contour winding numbers (Whitney 1937). In his thesis, S. Smale generalized this result ...

For any point P on the boundary of an ordinary ball, find a neighborhood of P in which the intersection with the ball's boundary cuts the neighborhood into two parts, each ...

A subset E of a topological space S is said to be of first category in S if E can be written as the countable union of subsets which are nowhere dense in S, i.e., if E is ...

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 ...

A property that passes from a topological space to every subspace with respect to the relative topology. Examples are first and second countability, metrizability, the ...

Let Y^X be the set of continuous mappings f:X->Y. Then the topological space Y^X supplied with the compact-open topology is called a mapping space, and if X=I is taken as the ...

In algebraic topology, a p-skeleton is a simplicial subcomplex of K that is the collection of all simplices of K of dimension at most p, denoted K^((p)). The graph obtained ...

A collection of subsets of a topological space that is contained in a basis of the topology and can be completed to a basis when adding all finite intersections of the ...

A characterization of normal spaces which states that a topological space X is normal iff, for any two nonempty closed disjoint subsets A, and B of X, there is a continuous ...