User:IssaRice/Analogous results between subfields of math

Finiteness vs compactness

see https://www.math.ucla.edu/~tao/preprints/compactness.pdf

considering domain of maps

If A is a set such that every continuous f: A -> R attains a max, then A is compact (?) (found in Schramm's analysis text).

This reminds me of the homomorphism characterization of normal subgroups. A subgroup H is normal iff there is some f : H -> G such that ker f = H.

There's also a similar characterization for connected sets, i.e. A is connected iff every continuous function f: A->R has the intermediate value property, which says that if f(a) < 0 and f(b) > 0 then there is c in [a,b] such that f(c)=0. (this only works on the real line?)

Injective iff surjective

• If A and B are finite sets that have the same cardinality, then $f:A\to B$ is injective iff surjective.
• If V and W are finite-dimensional vector spaces with the same dimension, then a linear map $T:V\to W$ is injective iff surjective.

See also

• https://machinelearning.subwiki.org/wiki/User:IssaRice/Subfield_of_math_as_study_of_concepts_preserved_under_transformation