Open and close set (Topology) I have an exercise about topology to do for tomorrow but I am really not sure of my answers or don't have any answer. Can you help me figure them out ? Thank you. Let E be a non-empty bounded set of real numbers and put α = supE, and β = inf E. Assume that α doesn't ∈ E and β ∈ doesn't E.Which of the following statements is true and which is false. In each case justify your answer. (a) E is an open set. (b) E is not a closed set. (c) E is an innite set. (d) (α, β) ⊂ E. My answers are : a) True since all points in E are interior points. b) True supE or infE may be a limit point for E. c) I don't know this. d)False E ⊂ (α, β). Is that correct? And what is the answer for c) ?

The answers are as follows :

a) False. Take $E = (0, 1]\cup [2,3)$. This satisfies the hypothesis, but is not open (1 is not an interior point)

b) True. $\sup(E)$ and $\inf(E)$ are limit points (For every $\epsilon > 0$, $\alpha - \epsilon$ is not an upper bound, so ..)

c) If $E$ is finite, then $\sup(E) = \max(E)$ and so must be in $E$.

d) True, as you say.