Kevin O'Brien

I am studying complex analysis and I need to understand conformal mappings — angle-preserving transformations of the complex plane. I know that any analytic function with non-zero derivative is conformal (preserves angles). The key examples are: - $w = z + a$ (translation) - $w = e^{i\theta} z$ (rotation) - $w = kz$ (scaling) - $w = 1/z$ (inversion) - $w = z^2$ (maps upper half-plane to whole plane minus a slit)

I am studying real analysis and I'm learning about the Baire Category Theorem. It states:

In a complete metric space, the intersection of countably many dense open sets is dense.