pi-base
Homogenous spaces
https://topology.pi-base.org/spaces?q=%3Fhomogenous
Spaces which are homogeneous: https://topology.pi-base.org/spaces/S000015 (obvious from definition) https://topology.pi-base.org/spaces/S000017 (obvious from definition) https://topology.pi-base.org/spaces/S000018 (product of homogeneous spaces) https://topology.pi-base.org/spaces/S000019 (obvious from definition since R is) https://topology.pi-base.org/spaces/S000032 (standard result, see e.g. Infinite-dimensional topology by van Mill) https://topology.pi-base.org/spaces/S000042 (obvious from definition)
Spaces which aren't homogeneous: https://topology.pi-base.org/spaces/S000038 (obviously not homogeneous because of the point $(0, 0)$) https://topology.pi-base.org/spaces/S000039 (not first countable only at $(\omega_1, 0)$ https://topology.pi-base.org/spaces/S000040 (not first countable only at $\infty$) https://topology.pi-base.org/spaces/S000041 (because $[0, 1]$ isn't) https://topology.pi-base.org/spaces/S000044 (the amount of open sets to which $x$ belongs must be the same for each $x$) https://topology.pi-base.org/spaces/S000045 ($0$ belongs to every open set, but there are points which don't) https://topology.pi-base.org/spaces/S000046 (if $a = 3/2$ and $b = 5/2$ then there are $U, V$ with $a\in U, b\notin U, a\notin V, b\in V$ yet no such neighbourhoods for $a = 1/2$ and $b = 1/4$)
This is just some of the spaces, but there is a lot of them which are easy to decide but have no mention on pi-base. Someone should update this.
