# Locally Euclidian, Hausdorff and Second Countable are independent conditions

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?

• @AguirreK, a two-point space $X$ with topology $\{\emptyset,X\}$. Commented Nov 22, 2016 at 16:42