Proving that the metric space $((0,\infty),d)$ is complete, with $d(x,y)=|\ln x-\ln y|$
51227
Proving that given metric space is complete: $X := (0,\infty)$ and $d:=|\ln(x)-\ln(y)|$
41298
Understanding proof of completeness of $L^{\infty}$
95783
Fixed points of contractions in metric spaces
90628
The metric spaces are Tychonoff spaces
69528
Metrizability of quotient spaces of metric spaces
85242
Deciding whether two metrics are topologically equivalent in the space $C^1([0,1])$
127034
Let $\rho(x,y)= \frac{d(x,y)}{1+d(x,y)}$, show that $\rho$ is a metric equivalent to $d$
146967
Understanding equivalent metric spaces
133152
(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.
151984
Proving the existence of a metric space
61202
Show that $C[a,b]$ is a complete space under the metric $d(f,g)=\sup_{t\in [a,b]}|f(t)-g(t)|$.
42638
Showing that a function $f$ has a unique fixed point in a metric space.
141643
Showing $\rho (x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric
98553
Infinite metric space has open set $U$ which is infinite and its complement is infinite
26547
Real numbers equipped with the metric $ d (x,y) = | \arctan(x) - \arctan(y)| $ is an incomplete metric space
52104
A pseudometric on the space of the measurable functions is complete
16060
Showing this metric space is complete