query: Prove that, for $x \in \mathbb R$ and $\delta_x > 0$, the open interval $(x-\delta_x, x+\delta_x)$ is itself an open set
positive_passages.docid: 135454
positive_passages.text: An open interval is an open set?
negative_passages.docid: 66239
negative_passages.text: Verifying that a certain collection of intervals of $\mathbb R$ forms a topology
negative_passages.docid: 138202
negative_passages.text: Every open set in $\mathbb{R}$ is a countable union of closed sets
negative_passages.docid: 73368
negative_passages.text: Does this proof that every open set in a metric space is a union of open balls use the axiom of choice?
negative_passages.docid: 21892
negative_passages.text: Any open subset of $\Bbb R$ is a countable union of disjoint open intervals
negative_passages.docid: 99464
negative_passages.text: Proof Concerning Indexed family of Sets and Intervals
negative_passages.docid: 78786
negative_passages.text: Why does an open interval NOT have measure zero?
negative_passages.docid: 73130
negative_passages.text: A simple question about open set
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: 143105
negative_passages.text: Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
negative_passages.docid: 123897
negative_passages.text: Sum of two open sets is open?
negative_passages.docid: 63442
negative_passages.text: Show that if $U$ is an open connected subspace of $\mathbb{R}^2$, then $U$ is path connected
negative_passages.docid: 148112
negative_passages.text: Prove that a topological space equipped with a delta-complex structure is Hausdorff
negative_passages.docid: 156520
negative_passages.text: Intervals are connected and the only connected sets in $\mathbb{R}$
negative_passages.docid: 60053
negative_passages.text: Does the boundary of an open set have measure zero (in $\mathbb{R}^n$)?
negative_passages.docid: 110957
negative_passages.text: No non-trivial clopen sets in $\mathbb{R}$? How to give a direct proof?
negative_passages.docid: 80827
negative_passages.text: Let $A$ be a nonempty subset of $\mathbb{R}$. Prove...