The question is, let $X$ be the union of the set $\mathbb{R} \setminus \{0\}$ and the two-point set $\{ p, q\}$. Topologize $X$ by taking as a basis the collection of all open interval in $R$ that do not contain 0, along with all the sets $f$ the form $(-a, 0) \cup \{p\} \cup (0,a)$ and all the sets of the form $(-a, 0) \cup \{q\} \cup (0,a)$, for $a>0$.

Prove that there is no continuous injection $X \to \mathbb{R^n}$ for any $n \in Z_+$.

I kind of get the intuition of this. Since it has two origins, there is no way to find a injective map. But I am struggling on how to write a formal proof by using this fact. (Or my intuition is actually wrong?)

Great thanks!

  • $\begingroup$ If you have a continuous injection $\iota \colon Y \hookrightarrow Z$, how does the original topology on $Y$ compare to the initial topology with respect to $\iota$? $\endgroup$ – Daniel Fischer Mar 8 '16 at 15:42
  • $\begingroup$ By Z, you mean R, right? $\endgroup$ – Xuan Mar 8 '16 at 15:45
  • $\begingroup$ I'm thinking much more general. We would of course apply it to the case $Z = \mathbb{R}^n$ here. $\endgroup$ – Daniel Fischer Mar 8 '16 at 15:46
  • $\begingroup$ Sry about that, but I do not really get this, can you explain a little bit more? Thanks a lot! $\endgroup$ – Xuan Mar 8 '16 at 15:50
  • $\begingroup$ How well acquainted are you with the concept of initial topologies? $\endgroup$ – Daniel Fischer Mar 8 '16 at 15:53

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