\times denotes the cartesian product (to my knowledge) universally. If 3rd-semester calculus is when you introduce a general definition of continuity (I am not from the US, wouldn't know how the programs usually work there) on either metric or topological spaces, the cartesian product starts to appear quite a lot I guess ?

Typically in the US, the calculus sequence is one semester differentiation, one semester integration, and a third semester of three dimensional and vector calculus. The × symbol is used a lot for vector cross products in the third semester. Typically these courses don't involve proofs. Serious students frequently take a portion of this sequence +/- matrix algebra in high school as AP courses or dual enrollment where the school cooperates with a local college to share their exams and get official credit. They are technically considered to be college level courses in the US. I think a lot of the content in them is covered in A level further maths or IB HL math or whatever your local equivalent is.

This sequence is followed by differential equations courses for the physicists, engineers, and most mathematics majors. Then every college has a mechanism to generate mathematical maturity in their first or second year pure math majors - sometimes it's a proof focused version of linear algebra, sometimes it's a specific Introduction to Proofs course, sometimes it's a discrete math/set theory course, sometimes it's groups/rings or real analysis but slowed down a bit at first. This gates the upper level pure mathematics courses, where most programs require one semester each of algebra and analysis and some number of elective courses.

A general definition of continuity typically doesn't arise until a topology course or a second semester real analysis course. It is entirely possible to graduate from most mathematics bachelor's programs in the US without taking either of those courses.

The \times sign is × in latex. To get • you use \cdot.

Yes, but it is rare to see \times used for scalar multiplication, isn't it?

In the US it's how we're taught as kids in gradeschool. We only switch to dot later on during algebra once "x" shows up as a variable.

IIRC at university-level we used the wedge symbol [0] for cross product - not sure how common that is.

https://en.wikipedia.org/wiki/Wedge_(symbol)

In my experience it would not be typical to use a wedge to represent a cross product. Typically a wedge is used to refer to the outer/exterior product, which in three dimensions would correspond to a bivector as opposed to the vector you get from a cross product.

Wikipedia says it’s more common in physics, and we mostly used it in that context (e.g. fluid mechanics) rather than pure math. It was pronounced “veck”, IIRC.

The × sign is also used for the Hartesian product, which comes up often when building up Lebesgue integration.