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 is injective iff surjective.
- If V and W are finite-dimensional vector spaces with the same dimension, then a linear map is injective iff surjective.
