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A topological space is compact if every open cover of X has a finite subcover. In other words, if X is the union of a family of open sets, there is a finite subfamily whose ...

Connectivity properties obey the following hierarchy: convex => star convex => pathwise-connected => connected.

Donaldson (1983) showed there exists an exotic smooth differential structure on R^4. Donaldson's result has been extended to there being precisely a continuum of ...

A branch of topology dealing with topological invariants of manifolds.

D^*Dpsi=del ^*del psi+1/4Rpsi, where D is the Dirac operator D:Gamma(S^+)->Gamma(S^-), del is the covariant derivative on spinors, and R is the scalar curvature.

D^*Dpsi=del ^*del psi+1/4Rpsi-1/2F_L^+(psi), where D is the Dirac operator D:Gamma(W^+)->Gamma(W^-), del is the covariant derivative on spinors, R is the scalar curvature, ...

Let Y^X be the set of continuous mappings f:X->Y. Then the topological space for Y^X supplied with a compact-open topology is called a mapping space.

A pointed space is a topological space X together with a choice of a basepoint x in X. The notation for a pointed space is (X,x). Maps between two pointed spaces must take ...

A Cartesian product equipped with a "product topology" is called a product space (or product topological space, or direct product).

A topological space having a countable dense subset. An example is the Euclidean space R^n with the Euclidean topology, since it has the rational lattice Q^n as a countable ...