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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 $\mathbb{S}^1$.

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