While reading Topology, J.Munkres (2014), there's a following statement:
"The subset $[a, b)$ of $\Bbb R$ is neither open nor closed."
To understand this, first I have think first,
1) Is $\Bbb R$ topology? Yes. It contains empty set and itself, while closed under infite unions of its element and finit intersection of its elements.
2) However, I think the set $[a,b)$ is in $\Bbb R$ so it's open. but it says neither one of open or closed.
This apparently looks I am missing something logically, but I am short of it.
Anyone help me to understand and good start with topology?
Best Answer
Assume that intervals of the form $[a,b)$ are open in $\mathbb{R}$ with $b\in\mathbb{R}$. Then since $\mathbb{R}$ is open (it is in fact both open and closed) then $\mathbb{R}\setminus[a,b)=(-\infty,a)\cup[b,+\infty)$ must be closed. But then $(-\infty,a)\cup[b,+\infty)$ can be expressed as the union of three sets $(-\infty,a)\cup[b,c)\cup(d,+\infty)$ with $b<d<c$ which is again open since all three sets are open. Therefore a contradiction. Next assume that intervals $[a,b)$ are closed with $b\in\mathbb{R}$. By definition any sequence $(x_n)\subseteq [a,b)$ if it converges its limit point is still in the interval $[a,b)$. Take $x_n:=b-1/n$ for $n\in\mathbb{N}$. You can convince yourself that for each $n$ it holds $x_n\in[a,b)$ but $\lim_nx_n=b\neq[a,b)$. Hence $[a,b)$ is not closed either.