pi-base
Space Suggestion: [newspacename]
Space Suggestion
$X = \lbrace 0 \times [0,1] \rbrace \bigcup \lbrace [0,1] \times 1 \rbrace \bigcup \lbrace \frac{1}{n} \times (0, \frac{1}{n}] \rbrace_\mathbb{N} \bigcup \lbrace [\frac{1}{n+1}, \frac{1}{n}] \times \frac{1}{n+1} \rbrace_\mathbb{N} \subset \mathbb{R}^2$
This space provides an example satisfying the search connected, locally connected, separable metric space that's $\sigma$-compact but not locally compact. It's also path-connected and locally path-connected. By replacing the segments with pseudo-arcs it loses path-connected and locally path-connected everywhere.
Actually, the harmonic comb with $(0,0)$ included but the vertical endpoints removed also works, that's easier.
FYI, when you create an issue, you are supposed to modify the title to reflect the content. I modified the title of #1501 for you, but you can do it here also.
That space does not seem locally path connected?
Wait, yes I see it now. It is locally path connected.
But the "harmonic comb" with $(0,0)$ included would not be locally connected or locally path connected, right?
pi-base currently does not have an example of path connected, metrizable, sigma-compact and non-locally compact space: https://topology.pi-base.org/spaces?q=37%2B53%2B17%2B%7Ewlc
How about this even simpler example: "comb with decreasing open tines": $X$ = the subspace of $\mathbb R^2$ consisting of the closed interval $[0,1]\times\{0\}$ together with all open vertical segments $\{\frac1n\}\times[0,\frac1n)$ for $n=1,2,\dots$.
pi-base currently does not have an example of path connected, metrizable, sigma-compact and non-locally compact space: https://topology.pi-base.org/spaces?q=37%2B53%2B17%2B%7Ewlc
In fact S75 is, as long as #990 gets merged.
#1234 is an another example satisfying “Path connected + Metrizable + σ-compact + !Locally compact”.
