I am having some trouble understanding what a topology means:
The definition says that: if $X$ is a non-empty set then a set $T$ of subsets of $X$ is said to be a topology on $X$ if
(i) $X$ and the empty set,$\emptyset$, belong to $T$
(ii) The union of any (finite or infinite ) number of sets in $T$ belongs to $T$
(iii)The intersection of of any two sets in $T$ belongs to $T$
Okay then if I define the sets $T_1$ and $T_2$ to be:
$T_1=\{[2,3],\emptyset,R\}$
$T_2=\{]-\infty,2[ \cup ]3,+\infty[,[2,3],\emptyset,R\}$
Then I guess that $T_1$ and $T_2$ must be two topologies on the set of real numbers $R$ as they satisfy the above three conditions.
Another definition says that the members of a topology are said to be open sets, so I want to ask if the set $[2,3]$ is considered to be open in the topological space ($R$,$T_1$)? and is this same set $[2,3]$ considered to be both open and closed in the toplological space ($R$,$T_2$) as it is present with its complement in $T_2$? and what about the subset $[5,7]$ of $R$ is it considered to be neither open nor closed in the above two topological spaces??
Any help is appreciated.
1) $[2,3]\in\tau_{1}$, so $[2,3]$ is open in $\tau_{1}$.
2) $[2,3]\in\tau_{2}$, so $[2,3]$ is open in $\tau_{2}$. Now $[2,3]^{c}=(-\infty,2)\cup(3,\infty)\in\tau_{2}$, so $[2,3]^{c}$ is open in $\tau_{2}$, so $[2,3]$ is closed in $\tau_{2}$.
3) $[5,7]\notin\tau_{1},\tau_{2}$, so $[5,7]$ is neither open in $\tau_{1}$ nor in $\tau_{2}$. $[5,7]^{c}=(-\infty,5)\cup(7,\infty)\notin\tau_{1},\tau_{2}$, so $[5,7]$ is neither closed in $\tau_{1}$ nor in $\tau_{2}$.