Subsets of $(X,d)$ where $d$ is the discrete metric are both closed and open
59654
Show that in a discrete metric space, every subset is both open and closed.
8015
Compact set plus a closed set is a closed set
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
123753
In a metric $(X,d)$, prove that for each subset $A$, $x\in\bar{A}$ if and only if $d(x,A)=0.$
143105
Show that, for every real number a, the set {$x ∈ S | f(x) ≤ a$} is a closed subset of $\mathbb R^d$
98553
Infinite metric space has open set $U$ which is infinite and its complement is infinite
19289
complement of a set and its measure
158155
Is set of isolated points of S closed?
136108
Arbitrary intersection of closed, connected subsets of a compact space connected?
48369
discrete subgroups of euclidean space
44623
Closed set as a countable intersection of open sets
79568
Cardinality of the set $D$
26547
Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space
144353
A closed ball in a metric space is closed
42698
Set of limit points of a subset of a Hausdorff space is closed.
64861
Intersection of open affines can be covered by open sets distinguished in *both*affines
9355
If $d(x,y)$ is a metric, then $\frac{d(x,y)}{1 + d(x,y)}$ is also a metric