Fundamental Branches of Mathematics

Question

I'm trying to drill down to the core branches of mathematics and it seems to me that Algebra, Analysis and Geometry are the fundamental branches of mathematics––every other 'branch' (topology, number theory, combinatorics, probability, statistics etc.) rest on these three in one shape or form. For example, vector analysis is the interaction between algebra (linear algebra, specifically) and analysis and when applied to (metric) spaces, links geometry. However, I cannot find anything to support this nor is my breadth of mathematical large enough to verify this. Is this perspective correct? I'm happy to be corrected on this.