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Throughout my reading I've encountered theorems which use certain wording, which is unclear at times. For example, consider the following corollary taken out of Intro to Topology by Mendelson,

Let $X$ and $Y$ be topological spaces, let $f:X\to Y$ be a continuous mapping of $X$ onto $Y$, and let $X$ be connected; then $Y$ is connected.

Does the fact that it's written "..mapping $X$ $\textbf{onto}$ $Y$" imply that the function $f$ is surjective, i.e., onto? I've also seen $X$ into $Y$ and I'm wondering if that means injective?

Thanks for any feedback!

S.D.
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  • I have never heard injectivity referred to as "into" – Sujaan Kunalan Jul 24 '13 at 05:51
  • Okay, how about for onto? Does that ever mean the function itself is onto? – S.D. Jul 24 '13 at 05:53
  • Sujaan, I accidentally deleted the comment you just made. Thank you for the answer, it helped clear things up. – S.D. Jul 24 '13 at 06:20
  • @ShantDanielian From what I was reading at wikiproof, a function f that maps, as in your example, X onto Y is actually surjective. On the other hand, if the function is not surjective you write that f maps X into Y. Not gonna lie, I was actually blown away. The link to the article is this: Wikiproof – Giordano Ribeiro Sep 20 '21 at 21:46

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Yes, a mapping of $X$ onto $Y$ is a surjection from $X$ onto $Y$.

When $f$ is described as a function from $X$ into $Y$, you cannot infer that it is injective. Usually it just means that $f:X\to Y$, i.e., that $Y$ is the codomain of $f$. Once in a great while you might see it used to mean that $f$ is a function from $X$ to $Y$ that is not surjective, but if that’s what the author intends, it should be made explicit.

Brian M. Scott
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