Intereting Posts

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On real line $(-\infty,+\infty)$ is open as well as closed. On extended real line $[-\infty,+\infty]$, is $(-\infty,+\infty)$ still a closed set? Thank you.

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The complement of $(-\infty,\infty)$ is $\{-\infty,\infty\}$. A finite set is closed (but not open) in the standard topology on $\mathbb{R}$ (and the extended real line inherits topological properties from $\mathbb{R}$). Thefore $(-\infty,\infty)$ is not closed, but it is open.

**EDIT:** More directly, a set in $\mathbb{R} \cup \{ \pm \infty\}$ is closed if and only if it contains all of its limit points. But we can construct many sequences in $(-\infty,\infty)$ whose limit is not in $(-\infty,\infty)$. Sequences like $\{ n | n \in \mathbb{N}\}$, $\{ x^2 | x \in \mathbb{R}\}$, and so on, as others have pointed out.

**Hint:** Is the complementary set open in the extended line?

The extended line is first-countable, so a set is closed if and only if it contains all the limit of convergent sequences in it. What about $x_n=n$? It is a sequence of elements from $(-\infty,\infty)$, and it is convergent in the extended line. Where does the limit lie?

The extended real line is got by adding the points $-\infty$ and $+\infty$ to $\mathbb{R}$.

The complement of $\mathbb{R}$ is $\{-\infty,\infty\}$, which is closed. The complement of $\{-\infty,+\infty\}$ is $\mathbb{R}$.

Since the complement of a closed set is open, it follows that $\mathbb{R}$ is open.

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