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The invariance of domain theorem states that if f:M->N is a one-to-one and continuous map between n-manifolds without boundary, then f is an open map.

Let M^n be an n-manifold and let F={F_alpha} denote a partition of M into disjoint pathwise-connected subsets. Then if F is a foliation of M, each F_alpha is called a leaf ...

An interval in which one endpoint is included but not the other. A half-closed interval is denoted [a,b) or (a,b] and is also called a half-open interval. The non-standard ...

If F is a family of more than n bounded closed convex sets in Euclidean n-space R^n, and if every H_n (where H_n is the Helly number) members of F have at least one point in ...

An inaccessible cardinal is a cardinal number which cannot be expressed in terms of a smaller number of smaller cardinals.

A lattice embedding is a one-to-one lattice homomorphism.

A topological space is locally connected at the point x if every neighborhood of x contains a connected open neighborhood. It is called locally connected if it is locally ...

A family of subsets of a topological space such that every point has a neighborhood which intersects only a finite number of them.

A space X is locally pathwise-connected if for every neighborhood around every point in X, there is a smaller, pathwise-connected neighborhood.

If a sphere is covered by three closed sets, then one of them must contain a pair of antipodal points.