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Epsilon proof of an open set.
This is an easy question, I know it but I’m not sure how to approach it.
The question is:
Prove that if $U$ is open in $\mathbb{R}^n$ and $A$ is an open subset of $U$ then $\overline{A}\subseteq U$.
I want to prove that $\overline{A}\subseteq U$ by contradiction.
I thought of, say, the open set
$$ A = [0,1)\cup(2,3)$$
but $A$ is not open and so I can’t use the definition of $\overline{A}$ as the largest open subset of $A$, because $A$ is not open.
Can anyone help?
A:
1) Consider the complement $U^c=\mathbb{R}^n \setminus U$ of $U$.
2) Observe that $A$ is an open subset of $U$, since it is the union of open sets.
Therefore $\mathbb{R}^n\setminus A$ is closed, since it is the union of two closed sets. So, $\overline{A}\subset U$
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Code:
$line=”select * from master.messages where from_id=0 and to_id=1 and sent_time between now() and 15-feb-2016 and ended=1″;
$result=mysql_query($line);
while($row=mysql_fetch_assoc($result)) {
$from=$row[‘from’];
$message=$row[‘message’];
}
it returns only one row.
A:
You’re very much mixing your mysql_* family functions, which are deprecated now, with mysqli_* functions which are recommended.
Use the prepare() and bind_param() functions to prepare statements with parameters, and use the mysqli functions for fetching data.
You should also do error checking with mysqli functions.
Recommended Reading
Correctly handling MySQL errors and prepared statements
Best way to do error handling
In this example, I do both error checking and proper escaping of your data before it’s sent to the database.
0) {
while ($row =
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