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I have a question about the definition of a homotopy between loops:

Let $\alpha$ and $\beta$ be loops with base point $x$ in a topological space $X$. A homotopy from $\alpha$ to $\beta$ is a continuous function $H$ from $[0,1]^2$ to $X$ such that $H(s,t)=f_s(t)$ where $f_s$ is a loop with base point $x$ and such that $f_0=\alpha$ and $f_1=\beta$.

What are the opens relative to $[0,1]^2$? If we're calling $H$ continuous, don't we need to know what topology on $[0,1]^2$ we're talking about? Is a certain topology implied?

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    @DylanWilson Actually, not only that all functions to $X\times Y$ are of this form, but that any pairs of continous functions $f:Z\to X,g:Z\to Y$ yields such a continuous function.2012-12-12

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It's just the usual topology, i.e. the product topology on $[0,1] \times [0,1]$.