Proving Inverse Function Continuity: A Topological Challenge

stgermaine
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Homework Statement


Prove that if the inverse of a function between topological spaces maps base sets to base sets, then the function is continuous.


Homework Equations


I have no idea.


The Attempt at a Solution


I seriously have no idea. This is for my analysis course, and I'm not sure why the prof is going over topology. I have some PDF's so I have some idea about open sets and continuous functions but not enough to solve this proof.
 
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Maybe start by giving the definition of a continuous function? What is it that you need to prove?

Also start by defining basis.
 

Thread 'Finding the nth roots of a complex number'

There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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