query: Let $X$ be a metric space $\mathcal{G}$ an open subset of $X$ and $F \subset \mathcal{G}$ finite. Show that $\mathcal{G} \setminus F$ is open.
positive_passages.docid: 148992
positive_passages.text: A finite set is closed
negative_passages.docid: 146485
negative_passages.text: Prove that the product space is a metric space.
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: 123039
negative_passages.text: If $0<\alpha < 1$ and $d(x,y)$ is metric, show that $((d(x,y))^{\alpha}$ is metric.
negative_passages.docid: 16654
negative_passages.text: Unions and Intersections of Open Sets are Open
negative_passages.docid: 108619
negative_passages.text: Proving Baire's theorem: The intersection of a sequence of dense open subsets of a complete metric space is nonempty
negative_passages.docid: 83159
negative_passages.text: Defining a metric space
negative_passages.docid: 21599
negative_passages.text: $(0,1)$ is an open subset of $\mathbb{R}$ but not of $\mathbb{R}^2$, when we think of $\mathbb{R}$ as the x-axis in $\mathbb{R}^2$. Prove this.
negative_passages.docid: 80860
negative_passages.text: How to call a space in which every G$_\kappa$-set is open
negative_passages.docid: 27855
negative_passages.text: Show that $(a,b)\times (c,d)$ is an open set in $\mathbb{R}^2$ with the Euclidian metric.
negative_passages.docid: 94825
negative_passages.text: Open set as a finite union of closed sets
negative_passages.docid: 38033
negative_passages.text: A surjective linear map into a finite dimensional space is open
negative_passages.docid: 28804
negative_passages.text: No. of possible dense subsets of a metric space
negative_passages.docid: 128894
negative_passages.text: About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$
negative_passages.docid: 67279
negative_passages.text: Is the union of finitely many open sets in an omega-cover contained within some member of the cover?
negative_passages.docid: 110217
negative_passages.text: $GL(n,K)$ is open in $M(n,K)$
negative_passages.docid: 113688
negative_passages.text: Is Topological Space a Metric Space?