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Continuum theory, point-set topology, spaces with algebraic structure, foundations, dimension theory, local and global properties.

5 votes

Examples where it's useful to know that a mathematical object belongs to some family of objects

There are $12$ elliptic plane cubics with fixed $j$ invariant passing through $8$ points in the plane ($4$ or $6$ if $j = 0, 1728$). Proof: Use the pencil of cubics argument to show that there are $ …