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If K is a simplicial complex, let V be the vertex set of K. Furthermore, let K be the collection of all subsets {a_0,...,a_n} of V such that the vertices a_0, ..., a_n span a ...

A characterization of normal spaces with respect to the definition given by Kelley (1955, p. 112) or Willard (1970, p. 99). It states that the topological space X is normal ...

If X is any space, then there is a CW-complex Y and a map f:Y->X inducing isomorphisms on all homotopy, homology, and cohomology groups.

The Chern number is defined in terms of the Chern class of a manifold as follows. For any collection Chern classes such that their cup product has the same dimension as the ...

Let X be a continuum (i.e., a compact connected metric space). Then X is hereditarily unicoherent provided that every subcontinuum of X is unicoherent. Any hereditarily ...

A homotopy from one embedding of a manifold M in N to another such that at every time, it is an embedding. The notion of isotopy is category independent, so notions of ...

Given a map f from a space X to a space Y and another map g from a space Z to a space Y, does there exist a map h from X to Z such that gh=f? If such a map h exists, then h ...

In abstract topology, a machine is method for producing infinite loop spaces and spectra. In automata theory, an abstract machine that is implemented in hardware is simply ...

A compact manifold admits a Lorentzian structure iff its Euler characteristic vanishes. Therefore, every noncompact manifold admits a Lorentzian structure.

If a compact manifold M has nonnegative Ricci curvature tensor, then its fundamental group has at most polynomial growth. On the other hand, if M has negative curvature, then ...