I know that Semi-locally Simply connectedness (SLSC) is strictly weaker than either one of Simply connectedness (SC) and Locally Simply Connectedness (LSC), with the latter two not necessarily implying each other ($S^1$ is LSC and not SC, the Cone of the Hawaiian earring is SC but not LSC).
I wanted to understand the SLSC property better, but the most revealing example that is given is the Hawaiian earrings, which is NOT SLSC, and so neither SC nor LSC. I can't seem to find an example of a space that is SLSC but neither SC nor LSC.
The examples I mentioned above are one of SC or LSC but not the other, so they are good examples for spaces that are SLSC and not SC or LSC, but I can't seem to put the ideas together to construct a space that is only SLSC.
Hope I'm not missing something obvious, Any examples are appreciated.
Take the wedge (one-point union) $S^1 \vee CH$ of the circle $S^1$ and the cone $CH$ of the Hawaiian earring.
More generally, take any LSC but not SC space X and any SC but not LSC space $Y$ and form their wedge.