If $f$ is a contraction map and $X$ is compact, show that there exists a fixed point
97914
Contractive mapping on compact space
39321
If $f:\mathbb R \to \mathbb R$ satisfies $f'(x) <L$ for some $L<1$, then $f$ has a unique fixed point
7081
isometry $f:X\to X$ is onto if $X$ is compact
126759
Unbounded continuous function on non-compact metric space
156850
Fix-Point Theorem Proof.
54729
Prove that $f$ has a fixed point .
95831
Compactness and sequential compactness in metric spaces
43376
If $A$ is compact and $B$ is closed, show $d(A,B)$ is achieved
119105
If $(X,d)$ is a complete metric space and $A$ is closed then show that for $x \in X$ there exists an element $a_0 \in A$ such that $d(x,A)=d(x,a_0)$
99645
Metrizable topological space $X$ with every admissible metric complete then $X$ is compact
21178
Show that $f$ extends to a smooth map.
11114
Is a closed subset of a compact set (which is a subset of a metric space $M$) compact?
27948
show that a function $A$ such that $ \rho (Ax,Ay)< \rho (x,y) $ $ \forall x\neq y $ not necessary has a fix point $ (Ax\neq x \space \forall x )$
26794
Metrizability of Hausdorff continuous image of compact metric space
19111
If every real-valued continuous function is bounded on $X$ (metric space), then $X$ is compact.
149999
fixed point of a continuous map on a projective space
149390
Fixed point of a Function (monotonically increasing)