Category: Math

There are multiple definitions of logical independence out there in the literature. In this article, we tease out these definitions and investigate their relationships with each other.

Here, we try to solve the following problem (and some related ones):

Problem. Prove that if such that and , then either or .

In multivariable calculus, we study line integrals and surface integrals, as well as volume integrals and various theorems involving these concepts, including for example the gradient and divergence theorems. We then learn that manifold calculus provides vast generalizations of these concepts and results. We try to understand and tease this out further in this article.

In general topology, a function is defined to be continuous if the preimages of open sets in are open in . This doesn’t define however functions that are continuous on just a subset of or at even a point. An application of continuity on subsets would be topological fields where the multiplicative inverse couldn’t possibly be continuous on all of , since includes 0. Since topological fields are standard and well-known concepts, the idea of generalized continuity on subsets must also be standard and well-known. In this article, I scour the Internet to understand how we define this today according to already-established standard literature.

There are multiple equivalent definitions of the (sub)ring generated by a subset. In this article, we state these definitions and prove their equivalence.

We want to define topological groups as having a compatibility axiom that intuitively says that the “group operation is continuous.” But the group operation is a multivariable function, so how would we define this? And what about for general topological algebras? (Here, “algebra” is meant in the universal algebra sense, not in the “vector space with bilinear multiplication” sense.)