# Some confusion regarding a quotient space: what is the space $\mathbb{S}^1\times I/ \mathbb{S}^1\times \{1\}$?

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 $$1$$.

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 $$x = y$$.