Let topological space $X$ and $f:X \to C$. Show that the space $X_{f}=\{g \in C(X) \mid \sup|g-f|<\infty\}$ is a complete metric space.
125610
Space of bounded continuous functions is complete
49374
Closure of a subset in a metric space
15480
Open set in a metric space is union of closed sets
146485
Prove that the product space is a metric space.
86970
Let $\{x_n\}$ be a sequence in a metric space such that $\{x_{2n}\}$, $\{x_{2n+1}\}$ and $\{x_{3n}\}$ all convergent. Show that $\{x_{n}\}$ converges
137881
A not complete metric space?
42638
Showing that a function $f$ has a unique fixed point in a metric space.
148403
Compact space and Hausdorff space
94438
Show that the countable product of metric spaces is metrizable
7402
Showing a compact metric space has a countable dense subset
80860
How to call a space in which every G$_\kappa$-set is open
86782
A $T_4$ space satisfying that 'every continuous map from it to a Hausdorff space is a closed map' is compact.
88185
Proof that a subspace $A$ of a complete metric space $X$ is complete iff $A$ is closed
144353
A closed ball in a metric space is closed
77563
Homeomorphism between spaces equipped with cofinite topologies
99645
Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
150987
Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent