0
$\begingroup$

Let $(X, \tau)$ be a topological space. Let $A\subset X$

My professor defined the closure of $A$ as the set of points in $X$ so that $\forall$ $U$(open) containing $x$, $U \cap A$ is non-empty.

The question I have is which of the following ones are logically equivalent to the one the Professor gave?

$\forall$ $U$(open) containing $x$, $U \cap A$ is non-empty.

(1) $\forall U$ ($ U \in \tau$ and $ x\in U \implies U \cap A $ is not empty)?

Or

(2) $\forall U$ ($ U \in \tau$ and $ x\in U \textbf{ and } U \cap A $ is not empty)?

Or

(3) $\forall U$ ($ U \in \tau$ $\implies$ $ x\in U \textbf{ and } U \cap A $ is not empty)?

Please explain why.

$\endgroup$
  • $\begingroup$ Yes, they are equivalent. $\endgroup$ – Kavi Rama Murthy Jul 9 at 0:29
  • $\begingroup$ @KaviRamaMurthy why can’t it be: $\forall U$ $( U \in \tau$ and $ x\in U \textbf{and} U \cap A $ is not empty)$ ? $\endgroup$ – topologicalmagician Jul 9 at 0:32
  • $\begingroup$ @topologicalmagician Because that would assert that, for every (not necessarily open) subset of $X$, that $U \in \tau$ (i.e. every subset is open) and $x \in U$ (every subset contains $x$) and $U \cap A \neq \emptyset$ (every subset intersects $A$). This is never true, as it doesn't exclude $U = \emptyset$. $\endgroup$ – Theo Bendit Jul 9 at 1:19
0
$\begingroup$

Only 1 is correct: when we quantify over all open sets (or equivalently over all $U \in \tau$)) we know the implication if $x \in U$ then $U$ must intersect $A$. We do not know that all open sets contain $x$; this will be exceedingly rare or even false as $\emptyset$ is open in any $X$..

So correct is

$$\forall U \in \tau: (x \in U \implies U \cap A \neq \emptyset)$$

$\endgroup$
1
$\begingroup$

1 is correct.

2 implies that all open open sets contain $x$.
That is false. For example $\Bbb R - \{x\}$ is open and excludes $x$.

3 implies that if $U$ is an open set, then $x \in U$
which again is false. $U=\emptyset$, for example.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.