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Consider $X=\mathbb{R}$ with the standard topology. How can I find a compactification $Y$, such that $Y$ is (of course) compact, hausdorff and has a subset which is homeomorphic to $\mathbb{R}^2$?

This question is a part of a bigger question, where in the first bullets I proved that $$ \left\{ \left(t,\sin\left(\frac{1}{1-t^{2}}\right)\right)\in\mathbb{R}^{2}\thinspace:\thinspace t\in\left(-1,1\right)\right\} \cup\left(\left\{ -1,1\right\} \times\left[-1,1\right]\right) $$

Is a compactification of $\mathbb{R}$. Can it be utilized in order to deduce a compactification as I stated above?

Thanks in advance.

FreeZe
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    You can use a similar idea to this example. Think of what's happening in this example as "the function $f(t)$ oscillates all around the range $[-1, 1]$ as $t \to 1$, so the closure of the graph ${(t, f(t)) : t \in (-1, 1)}$ contains ${1} \times [-1, 1]$". Now try modifying this idea, but with an extra dimension: think about a set like ${(t, f(t), g(t)): t \in (-1, 1)}$, and try to make it so that the closure of this set contains ${1} \times [-1, 1]^2$... – Izaak van Dongen Jul 13 '24 at 13:47
  • It is no duplicate of https://math.stackexchange.com/q/4610897, but it contains the answer to your question. – Paul Frost Jul 30 '24 at 15:40

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