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I am reading through a book on topology and came across the following phrase: for each element $x\in X$ we can uniquely define a function $f_x: A\rightarrow B$ such that $f_x$ satisfies some topological property $P$.

What exactly does this mean? Does this mean that for each $x\in X$ there is only one $f_x$ satisfying the property $P$? Or does it mean that if you have different elements $x\neq y$ in $X$ then $f_x\neq f_y$?

The terminology is a bit confusing to me.

fosho
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    What book are you referring to? What page? What is the context? How was "topological property" defined in the text? What is $A$, and how does it relate to $X$? – Card_Trick Dec 18 '18 at 22:29
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    Without the context suggested by @Card_Trick, it is hard to say. However, I would interpret what you have written to mean "For each $x$, we can define a function $f_x$. For any fixed $x$, this function is uniquely defined." That is, once we fix the value of $x$, the function corresponding to that value is unique. – Xander Henderson Dec 18 '18 at 22:32

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