TITLE: which of the spaces are Locally Compact
QUESTION [7 upvotes]: [NBHM_2006_PhD Screening Test_Topology]

which of the spaces are Locally Compact

$A=\{(x,y): x,y \text{ odd integers}\}$
$B=\{(x,y): x,y\text{ irrationals}\}$
$C=\{(x,y): 0\le x<1, 0<y\le 1\}$
$D=\{(x,y): x^2+103xy+7y^2>5\}$


A topological space $X$  is locally compact if every point has a neighborhood which is contained in a compact set. 
well, I can prove that $\mathbb{Q}$ is not locally compact, so 1,2, are not Locally Compact, 3 is clearly locally compact. I am not ssure about 4. thank you.

REPLY [5 votes]: For 4): 
All open or closed subsets of a locally compact Hausdorff space are locally compact in the subspace topology. $R^2$ is locally compact and Hausdorff and $D = p^{-1}((5, \infty))$ is the inverse image of an open set under a continuous function $p(x,y) = x^2+103xy+7y^2$.