I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to self study. Besides some basic topological definitions, are there specific areas where I should become acquainted with? I have heard multivariable calculus is a 'prerequisite' for the study of smooth manifolds -- is this from an intuition perspective, and if so, what parts and to what degree of rigor are they referring?
Thank you. (My apologies if this is a repeat question, by the way -- I could not find it)