# Topology associated to winding number

Let $$X$$ be the space of continuous functions $$\mathbb{S}^1\to\mathbb{C}\setminus\{0\}$$.

We define a map $$w:X\to\mathbb{Z}$$ as the winding number, or degree of such a map: $$w(f) := \frac{1}{2\pi i}\int_{\mathbb{S^1}}\frac{f'(z)}{f(Z)}dz$$. At first this is only defined for smooth maps and then extended to continuous maps.

What is the initial topology on $$X$$ with respect to the map $$w:X\to\mathbb{Z}$$ (where $$\mathbb{Z}$$ is understood with the discrete topology)? Is it induced by a norm? or a metric?

In defining this notion of close-ness for two functions, it seems to me like the situation is qualitatively different than that of functions $$f:\mathbb{S}^1\to\mathbb{S}^1$$, since here if two functions get arbitrarily close to zero on some point of $$\mathbb{S}^1$$, they could be very close together in the sense that they take the same value for an arbitrarily big part of $$\mathbb{S}^1$$ yet they could change winding by passing through zero. So intuitively it seems like a measure of distance between $$f$$ and $$g$$ should explode as either $$f$$ or $$g$$ get close to zero at any point of $$\mathbb{S}^1$$.