query: closed ε-ball around x
positive_passages.docid: 144353
positive_passages.text: A closed ball in a metric space is closed
negative_passages.docid: 9320
negative_passages.text: Extension of continuous function on a closed subset of $\mathbb{R}$
negative_passages.docid: 53132
negative_passages.text: A convex subset of a set has 'smaller' boundary than the set?
negative_passages.docid: 115272
negative_passages.text: Showing that projections $\mathbb{R}^2 \to \mathbb{R}$ are not closed
negative_passages.docid: 97761
negative_passages.text: Show that a proper continuous map from $X$ to locally compact $Y$ is closed
negative_passages.docid: 150468
negative_passages.text: Example of two open balls such that the one with the smaller radius contains the one with the larger radius.
negative_passages.docid: 112303
negative_passages.text: Distances between closed sets on metric spaces
negative_passages.docid: 147914
negative_passages.text: Closed image of locally compact space
negative_passages.docid: 70959
negative_passages.text: Prove that a set $E$ is closed iff it's complement $E^{c}$ is open
negative_passages.docid: 47975
negative_passages.text: Closed cone in the euclidean space $\mathbb{R}^n$
negative_passages.docid: 12467
negative_passages.text: Why does this proof work: Closed unit ball in $C_0$ is not compact
negative_passages.docid: 114795
negative_passages.text: Closed form for the area of a convex cyclic n-gon, given the set of edge lengths
negative_passages.docid: 133152
negative_passages.text: (X,d) be a compact metric space. For every open cover, show there exists ε > 0 such that ∀𝑥 ∈ X, B(x,ε) is contained in some member of the cover.
negative_passages.docid: 149820
negative_passages.text: How do you prove the set $G = \{(x, f(x)) \mid x \in \mathbb R\}$ is closed?
negative_passages.docid: 22558
negative_passages.text: Vitali-type set with given outer measure
negative_passages.docid: 47639
negative_passages.text: Uncountable Cartesian product of closed interval
negative_passages.docid: 152193
negative_passages.text: The intersection of a closure with an open set