If I assume that my topology is defined on the extended real-valued numbers, then $\mathbb{R}\cup\left\{-\infty,+\infty\right\}=\left[-\infty,+\infty\right]$, acting as my entire space, is both open and closed.
Consider now the set $\left(\alpha,+\infty\right]$. Am I right to say that because its complement $\left[-\infty,\alpha\right]$ is certainly not open (hence closed), that $\left(\alpha,+\infty\right]$ is open? Or are both $\left[-\infty,\alpha\right]$ and $\left(\alpha,+\infty\right]$ neither open nor closed? Or, are they both open and closed?
My question might appear basic, but I am a bit puzzled here…
Best Answer
$(a,+\infty]$ is open because it contains an open neighbourhood of each of its elements. In fact, it is by definition of the topology an open neighbourhood of $+\infty$. Hence $[-\infty,a]$ is closed.