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If I am examining two sets $X$ and $Y$, each with the discrete topology, will $X\times Y$ have a discrete topology? My understanding is yes. I believe this because $X\times Y$ is the finite product of discrete spaces. Every point in $X$ is open and every point in $Y$ is open, and every point in $X\times Y$ is open. Thus $X\times Y$ has a discrete topology.

Is this understanding correct?

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    Yes. In fact, a product of $A$ discrete spaces, with the product topology, each with more than one point is discrete if and only if $A$ is finite.2011-12-27
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    You haven't really explained your argument. Why is it relevant that $X \times Y$ is the finite product of discrete spaces? *Why* is every point in $X \times Y$ open?2011-12-27
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    So, fundamentally, you are correct that to prove a topology on a set $Z$ is discrete, you only need to prove that singletons are open. Why is this enough? Because every subset of $Z$ is a union of singletons, and so if every singleton is open, then so is every subset of $Z$. But, as @ChrisEagle pointed out, you have to prove that singletons in $X\times Y$ are open - you can't just assert it. It depends on your definition of the product topology, however.2011-12-27

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