In any metric space X, the sets ∅ and X are always closed.
49721
Prove the empty set is closed for a metric space (X,d).
22240
Every infinite subset of a countable set is countable.
14219
Is the intersection of a closed set and a compact set always compact?
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
15480
Open set in a metric space is union of closed sets
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
26563
Help completing a proof regarding an infinite family of nested closed subsets in a compact metric space
10472
Is every set a subset?
36237
Closed set mapped to itself in a compact Hausdorff space
132641
Proving that Euclidean Space is a complete metric space
109638
Show that a complete metric space without isolated points is uncountable
114608
Proof concerning regular space: there exist a closed set contained in any open set containing $x$
123972
Proving that every function defined on a discrete metric space is uniformly continuous.
80860
How to call a space in which every G$_\kappa$-set is open
83159
Defining a metric space
124179
Proof that every metric space is normal
125580
Proving that every countable metric space is disconnected?