Some properties about k-topology I have regarding the $K$-topology which generated by $(a,b)$ and $(a,b)/K$ Where $a,b\in {R}$. Where $K=\{1/n: n=1,2,3,....\}$ This is what I know about it 
1) connected space 
2) not path connected 
3) finer than $R$ with usual topology 
4)not compact
5)not regular , not normal space as well 
6)second  countbale, so it is first countable, Lindelöf 
Separable 
7) not comparable with $R_L$ lower limit topology 
8) not metrizable space 
9) it is not countably compact, not limit point compact, not sequentially compact. 
Here is my question 
Is it locally connected?
Is it locally compact ? 
My answer is No for both, but I am entirely sure 
Any help will be appreciated 
 A: Since $\Bbb R$ with the usual topology is locally compact, and the only point at which the $K$-topology differs from the usual one is $0$, the point $0$ is the place to look for bad things to happen locally.
Let $\tau$ the $K$-topology. For local connectedness, suppose that $\mathscr{B}$ is a local $\tau$-base at $0$ whose members are all connected. let $U=(-1,1)\setminus K\in\tau$; by hypothesis there is a $B\in\mathscr{B}$ such that $B\subseteq U$. Since $B$ is a $\tau$-nbhd of $0$, there are an $a<0$ and $b>0$ such that $(a,b)\setminus K\subseteq B$. Choose $n\in\Bbb Z^+$ such that $\frac1n<b$; then $\left(\frac1{n+1},\frac1n\right)$ and $B\setminus\left(\frac1{n+1},\frac1n\right)$ are disjoint $\tau$-open subsets of $B$ whose union is $B$, contradicting the connectedness of $B$.
For local compactness, suppose that $U$ is a $\tau$-open nbhd of $0$. Then there are $a<0$ and $b>0$ such that $0\in(a,b)\setminus K\subseteq U$. Let $B=(a,b)\setminus K$. If $\operatorname{cl}_\tau U$ were $\tau$-compact, $\operatorname{cl}_\tau B$ would also be $\tau$-compact. (Why?) But $\operatorname{cl}_\tau B=[a,b]$, and $[a,b]\cap K$ is an infinite, closed, discrete subset of $[a,b]$, so $[a,b]$ isn’t even limit point compact, let alone compact.
