Let $X$ be compact and suppose $f:X\to X$ satisfies $d(f(x),f(y))<d(x,y)$ whenever $x\neq y$. Show that $f$ has a fixed point.
42638
Showing that a function $f$ has a unique fixed point in a metric space.
150379
Perfect Map $p:\ X\to Y$, $Y$ compact implies $X$ compact
148266
Let $f:X\to X $ be continuous. Does $f $ have a fixed point when $X=[0,1)$ or $X=(0,1) $?
55749
Compactly supported continuous function is uniformly continuous
30455
Show that any continuous $f:[0,1] \rightarrow [0,1]$ has a fixed point $\zeta$
82333
let $(X,d)$ be a metric space. How I can show that any finite subset of $X$ is closed.
98609
Does pointwise equicontinuous and uniformly equicontinuous implies compactness?
4171
If $f$ is locally Lipschitz on $X$ and $X$ is compact, then $f$ is Lipschitz on $X$.
18950
Show that $f$ has at most one fixed point
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.
34077
Let $f:\mathbb{R}\to\mathbb{R}$ satisfy $f\big(x-f(y)\big)=f\big(f(y)\big)+x\cdot f(y)+f(x)-1$ for all $x,y\in\mathbb{R}$.
126842
Fixed point in a continuous function
64694
$C_{c}(X)$ is complete. then implies that $X$ is compact.
156850
Fix-Point Theorem Proof.
99179
Continuous function with compact domain has continuous inverse
72856
Show $f$ is constant if $|f(x)-f(y)|\leq (x-y)^2$.
15579
Show that $f$ is uniformly continuous if limit exists