Locally Euclidian, Hausdorff and Second Countable are independent conditions

Question

When defining a smooth manifold $M$, one usually requires $M$ to be a topological space satisfying these $3$ conditions:

1) Locally euclidean

2) Hausdorff

3) Second countable

I want to verify that these are independent conditions, i.e., that one can find a topological space for each combination:

a) Locally Euclidean + not Hausdorff + not second countable

b) Not locally Euclidean + Hausdorff + not second countable (Ex.: $\mathbb{R}$ with coarse topology)

c) Not locally Euclidean + not Hausdorff + second countable

d) Locally Euclidean + Hausdorff + not second countable (Ex.: the long line $\omega_1\times[0,1)\setminus\{(0,0)\}$)

e) Locally Euclidean + not Hausdorff + second countable

f) Not locally Euclidean + Hausdorff + second countable (Ex.: union of two concurrent lines in $\mathbb{R}^2$)

I'm still stuggling with a), c) and e), since non-Hausdorff spaces are always so hard to think about... Any ideas?

Answer 1

For (e), take the real line with the doubled origin. (That is, it has two origins, and an open set around either origin must contain that origin and an interval around the origin, but not necessarily the other origin.)

For (a), take an example like the previous one, but use the long line instead.

For (c), you could do "basically anything", because locally Euclidean is easily the "strongest" of the conditions. E.g., take the indiscrete space on 2 points.