# Exercise with graph mapping: open set?

Consider a locally-bounded, outer semicontinuous set-valued mapping $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^m$ and an open, bounded set $O \subset \mathbb{R}^n$.

Say if the set

$$\bar{O} := \{ (x,y) \in \mathbb{R}^n \times \mathbb{R}^m \mid x \in O, (x,y) \in \text{graph}(F) \}$$

is open.

Note that $F$ being outer semicontinuous implies that $\text{graph}(F)$ is closed.

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Your accept rate is rather low. For instance, your last two questions were both quickly resolved $4$ days ago, but you've neither accepted the answers nor commented on them to point out what you're missing. Also, since you're asking a lot of questions about outer semicontinuous set-valued mappings, you might want to provide some motivation for them and perhaps explain how they're related. –  joriki Sep 12 '12 at 8:57
I accepted the last answers. I don't think my questions are related each other, sorry! –  Adam Sep 12 '12 at 9:06
Can we write the set as $\bar{O} = ( O \times \mathbb{R}^m ) \cap \text{graph}(F)$? –  Adam Sep 13 '12 at 6:42
Yes, we can.  –  joriki Sep 13 '12 at 7:33
Sorry, I meant to write $\{(x,y)\mid x\gt0\land y\gt0\land xy\ge1\}$. –  joriki Sep 13 '12 at 9:32