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What is meant by a single set in a topological space? The statement goes as: "let $X$ and $X'$ denote a single set in the topologies $\mathcal{T}$ and $\mathcal{T'}$ respectively".

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    I think it is meant as "$X$ and $X'$ are two topological spaces having the same underlying set". – 2017-02-20
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    Right, I think they mean for example that $X$ and $X'$ are, say, the unit interval in the reals with the usual and discrete topologies, respectively. Perhaps I would have worded it more clearly, it's a little confusing. – 2017-02-20
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    Ok,so by a "single set" what is meant is the same set but with two different topologies.. thanks! – 2017-02-20
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    Where did you see the term? Say if it is a book, then just check the word within the book :). – 2017-02-20
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    It is sometimes used to mean that the two topologies contain the same set, which is called $X$ when considered a member of $T$ and $X'$ when considered a member of $T'$. If memory serves, Munkres does that, but I don't have my copy of his intro to topology handy to check and I may be wrong. – 2017-02-20
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    I found it in Munkres. – 2017-02-21

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Here means,for example,a $\mathbf{one}$ set,say, $X_{0}$ endowed with two different topologies $\mathcal{T}$ and $\mathcal{T}'$, because in views of their different topologies,a $\mathbf{single}$ set $X_{0}$ could be two different topological spaces, namely $X$ and $X'$ respectively.But as sets,$X=X'$.