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I was reading a proof when I came upon a statement:

Suppose $f: S^{1} \rightarrow X .$ By hypothesis, there's a homotopy $h: S^{1} \times I \rightarrow$ $X$ from $f$ to a constant map. That is, $h_{0}=f$ and there is a point $x \in X$ such that, for all $s \in S^{1}, h(s, 1)=x .$ Because of the latter condition, $h$ factors through the quotient $(S^{1} \times I)/ (S^{1} \times \{1\}).$ That is, $h$ is equal to a composition $$ S^{1} \times I \rightarrow S^{1} \times I / S^{1} \times \{1\} \rightarrow X $$ where the first map is the quotient map.

I'm really confused about the convention used here, as I am used to a quotient space being defined as $X/\sim$ where $\sim$ is some equivalence relation defined on the space (set) $X$. So when they have written $(S^{1} \times I)/ (S^{1} \times\{1\})$, how are they defining the quotient space, which equivalence relation are they using?

Any help will be much appreciated, thank you!

The set $S^1$ is the circle in dimension $1$.

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Often in topology, one takes quotients by a subspace; this just means taking all the points in the subspace and identifying them to a single point. In your definition, this means $X / Y$ where $Y \subset X$ is any subspace is the quotient space $X / {\sim}$ where $x \sim y$ if $x$ and $y$ are in $Y$ or $x = y$.

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