Path connected space that is not locally connected

I am trying to construct a topological space that is path connected, but nowhere locally connected.

If we define $p = (0,1)\in\mathbb{R}^2$, and for all $q\in\mathbb{Q}\cap[0,1]$ $T_q = \{v\in\mathbb{R}^2\ |\ \exists t\in[0,1],\ v = (1-t)p + t(q,0)\}$, then $$ T = \bigcup\limits_{q\in\mathbb{Q}\cap[0,1]}T_q $$ the space of all lines connecting rationals in $[0,1]\times\{0\}$ to $p$, can be shown to be path connected but locally connected only at $p$. I thought of attempts to maybe generalize the idea to make the desired result. The immediate one of these attempts was to maybe 'rational duplicate' $p$ along the $y = 1$ line, i.e. to let $T$ be all the lines connecting every rational point on $[0,1]\times\{0\}$ to every rational point on $[0,1]\times\{1\}$, but the proof attempt did not go (I am not sure if this is correct).

Another idea was maybe a space-filling curve, but this seems a bit exotic.

In $\mathbb R^2$, draw lines from $(0,0)$ to $(1,q)$ for all rational $q$ in $[0,1]$
and from $(1,0)$ to $(0,-q)$ for all rational $q$ in $[0,1].$ This subspace
is path connected and nowhere locally connected.