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$(X, \tau) $ be a topological space.

$A\subset X$ is said to be first category (meager) if it can be expressed as a countable union of nowhere dense sets. Otherwise we call the set $A$ second category (co-meager).

So we divide subsets of $X$ in two different classes first category and second category.

Is the term "category" in Category theory entirely different from the category in topological spaces?

If yes, then what is the reason to use same word "category"in two different context ?

"Mathematics is the art of giving the same name to different things."-Henri Poincare .

Is the coincidence of the term " Category " a justification of Poincare's quote?

I don't think so. There is something I don't known but willing to know.

I know this question is slightly off topic and may be closed soon. But it's my interest to know your valuable insight.

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    $\begingroup$ They have strictly nothing in common, it is a pure linguistic coincidence, it turns out the word "category" is a common word, and therefore might be used by different people when they want to give a name to something. $\endgroup$ May 22 at 18:48
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    $\begingroup$ I used to wonder the same, but I only knew the category theory definition until today. After reading the definition in your post I can say that those two uses of the same word are as unrelated to each other as they are to its use to indicate the strength of a hurricane. $\endgroup$ May 23 at 1:10

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Perhaps disappointingly, there is no connection between the two uses at all. (Given this, I prefer to use "meager" for "first category," and my impression is that this is pretty common now.)

Note that the grammatical roles of the term are quite different - topological category refers to place in a (very simple) taxonomy, while a category-theoretic category is an object in its own right - so the terminological overload doesn't have much potential for confusion.

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