In the 4-dimensional Minkowski spacetime, for a given point $x = (x^0,x^1,x^2,x^3)$, its timelike future/past set is defined as, $$ I^{\pm}(x) = \{y =(y^0,...,y^3) \in \mathbb{R}^4 : \eta_{\mu \nu}(y-x)^{\mu}(y-x)^{\nu} > 0, \pm(y^0-x^0) >0 \} $$ where $(\eta_{\mu \nu}) = diag(1,-1,-1,-1)$, and one also defines the double cone sets as $ \mathcal{O}(x,y) = I^+ (x) \cap I^- (y).$

I have to show that $\mathcal{O}(x,y)$ is non-empty if and only if $y \in I^+(x)$ and $\mathcal{O}(x,y)$ is an open subset of $\mathbb{R}^4$.

I know I have to use the Hausdorff property, and the proofs are quite simple. But what I don't get is why does $\mathcal{O}(x,y)$ have to be open subset of $\mathbb{R}^4$, as the Hausdorff property definition doesn't say anything about the openness of the two subsets.

  • $\begingroup$ I would guess that you're parsing the question wrong. You have to show $O(x,y)$ is nonempty iff $y\in I^+(x)$. THEN you have to show $O(x,y)$ is always open (note that empty sets are trivially open). $\endgroup$ – Jahan Claes Apr 15 '19 at 21:14
  • $\begingroup$ You should show that [$\mathcal{O}(x,y)$ is non-empty if and only if $y \in I^+(x)$] and [$\mathcal{O}(x,y)$ is an open subset of $\mathbb{R}^4$]. $\endgroup$ – Jahan Claes Apr 15 '19 at 21:15
  • $\begingroup$ Yes, that did cross my mind. I did the two proofs independently as well, but the way the question was phrased made me wonder about some correlations between the two properties. $\endgroup$ – AphelionVoid Apr 15 '19 at 21:26
  • $\begingroup$ No, there's no correlation, because $O(x,y)$ is always open, so it can't figure in to a nontrivial iff statement. $\endgroup$ – Jahan Claes Apr 15 '19 at 21:43

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