Some topological spaces can be endowed with this property by adding just a single point at infinity in an Alexandroff extension. For 10 points each:
[10e] Name this property. A topological space has this nice property if every open cover of a set has a finite subcover.
ANSWER: compact [accept word forms like compactness; accept one-point compactification or Stone-Čech compactification]
[10m] The one-point compactification of the complex plane is this space, which is equivalent to the complex projective line. Points in this space are mapped onto points of the complex plane via stereographic projection.
ANSWER: Riemann sphere [prompt on sphere; prompt on Riemann manifold]
[10h] The analog of one-point compactification in non-commutative topology is the unitization of these spaces, which are dual to locally compact Hausdorff spaces by the Gelfand–Naimark theorem. The canonical example of one of these spaces is the set of bounded operators on a Hilbert space.
ANSWER: C-star algebras [prompt on star algebras or B-star algebras or Banach algebras; prompt on von Neumann algebra by asking “Can you be less specific?”]
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