I'm a graduate student in mathematics whose taking a topology class. The class recently started and I've been studying like crazy trying to understand, adapt and learn methods that work for writing proofs (and doing all the exercises I can and rewriting them over and over) but none of my efforts seem to be enough, so I'm turning to here in hopes if someone explains it to me, maybe I'll improve.
Let $R$$_{[1,2]}$ denote the subspace topology on [1,2] induced by the topology $R$ = { V $\subset$ $\mathbb{R}$ $\vert$ if x $\in$ V, then $\exists$ a,b $\in$ $\mathbb{R}$ a $<$ b such that x $\in$ (a,b] $\subset$ V}
The following is what I formally wrote.
Claim. A = [1,2] is $R$$_{[1,2]}$ closed.
Proof. The complement of A on the topology R$_{[1,2]}$ is [1,2]$\setminus$[1,2] = $\emptyset$ which $R$$_{[1,2]}$ open (as well as $R$-open because vaciously, $\emptyset$ $\in$ $R$ )
Claim. A = [1,2] is R$_{[1,2]}$ open.
Proof. Let [1,2] = (0,2] $\cap$ [1,2]. Since (0,2] is $R$-open, when intersected with [1,2], gives us a R$_{[1,2]}$ open set by definition.
Claim. B = {2} is not R$_{[1,2]}$ closed.
Proof. The complement of B in $R$$_{[1,2]}$ is [1,2) which is not open. Let x $\in$ [1,2), if x = 1, then x $\not$$\in$ (1,2] $\in$ $R$. Hence the complement of B in $R$$_{[1,2]}$ is not $R$-open.
Claim. B = {2} is not $R$$_{[1,2]}$ open.
Proof. Let x $\in$ {2}. Assume it is $R$$_{[1,2]}$ open, then there exists a,b $\in$ $\mathbb{R}$ such that x $\in$ (a,b] $\cap$ [1,2] = {2}. Then 2 $\leq$ max (a,1) < x $\leq$ min (b,2) $\leq$ 2, but that implies 2 $<$ 2 which is a contradiction, hence B is not $R$$_{[1,2]}$ open.
Claim. C = [1, $\frac{3}{2}$] is $R$$_{[1,2]}$ closed.
Proof. The complement of C in $R$$_{[1,2]}$ is ($\frac{3}{2}$,2] is $R$$_{[1,2]}$ open.
Claim. C = [1, $\frac{3}{2}$] is $R$$_{[1,2]}$ open.
Proof. Let [1, $\frac{3}{2}$] = (-2, $\frac{3}{2}$] $\cap$ [1,2] which is $R$$_{[1,2]}$ open because (-2, $\frac{3}{2}$] (is an $R$-open set) intersected with [1,2] gives us an $R$$_{[1,2]}$ open set by definition.
This is only one set of questions I answered. Please let me know and thank you.