query: Why only finite intersection of open sets is open
positive_passages.docid: 146261
positive_passages.text: Difference between topology and sigma-algebra axioms.
negative_passages.docid: 15480
negative_passages.text: Open set in a metric space is union of closed sets
negative_passages.docid: 11405
negative_passages.text: If a nonempty set of real numbers is open and closed, is it $\mathbb{R}$? Why/Why not?
negative_passages.docid: 44673
negative_passages.text: $M$ connected $\iff$ $M$ and $\emptyset$ are the only subsets of $M$ open and closed at the same time
negative_passages.docid: 21892
negative_passages.text: Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
negative_passages.docid: 46121
negative_passages.text: Does the intersection of two finite index subgroups have finite index?
negative_passages.docid: 136108
negative_passages.text: Arbitrary intersection of closed, connected subsets of a compact space connected?
negative_passages.docid: 42145
negative_passages.text: Prove that the only sets in $R$ which are both open and closed are the empty set and $R$ itself.
negative_passages.docid: 148507
negative_passages.text: Why an open set in $R^n$ might not be written as countable union of disjoint open intervals?
negative_passages.docid: 30079
negative_passages.text: Is for open connected $U$ the set $U_\varepsilon$ for small $\varepsilon$ connected?
negative_passages.docid: 75829
negative_passages.text: Dimension of open subsets of $R^n$
negative_passages.docid: 13132
negative_passages.text: If the union of two sets is contained in the intersection, then one is contained in the other ($[A\cup B\subseteq A\cap B]\implies A \subseteq B$)
negative_passages.docid: 101504
negative_passages.text: An open ball is an open set
negative_passages.docid: 74796
negative_passages.text: Extension of intersection of ideals
negative_passages.docid: 70538
negative_passages.text: Intersection and union of simply connected domain
negative_passages.docid: 106455
negative_passages.text: intersection of infinite collection of finite sets?
negative_passages.docid: 69283
negative_passages.text: The limit points of an open interval (open set)