If $X, d$ is a connected metric with at least two points, prove it is uncountable.
11455
Connected metric spaces with at least 2 points are uncountable.
121359
Show that $d_b(x,y)=\frac{d(x,y)}{1+d(x,y)}$ is a metric.
114770
Proof that a perfect set is uncountable
94438
Show that the countable product of metric spaces is metrizable
155777
Show that infinite Cartesian product is uncountable
123039
If $0<\alpha < 1$ and $d(x,y)$ is metric, show that $((d(x,y))^{\alpha}$ is metric.
149980
Are there any countable Hausdorff connected spaces?
57943
If $G$ is a convex set, prove that $\mathbb{C}\cup\{\infty\}\setminus G$ is connected.
83163
If $X$ is a connected metric space, then a locally constant function $f: X \to $ M, $M $ a metric space, is constant
99599
Uncountable and countable
82578
Show that if $G$ is $2-$connected, then for any two vertices $u$ and $v$ there exists a cycle $C$ such that $u, v \in V (C)$
12104
How to prove this result involving the quotient maps and connectedness?
130537
Proving that $d(f,g)=\|f-g\| = \sup \limits_{0\leq x \leq 1} |f(x)-g(x)|$ is a metric on $X=C[0,1]$
116494
What happens to the connectedness of $\mathbb R^2$ when countable many points are removed?
86685
Example of a set $S$ that is countable, but the set of limit points is uncountable
3649
Closed unit interval is connected proof
128894
About compactness and completeness of set $X = \mathbb{R}$ with the metric $d(x, y) = \frac{|x-y|}{1+|x-y|}$