Let $K$ be connected open set and $cl(K)$ compact such that $cl(K)\subseteq D$ where $D$ connected open set in set of complex numbers and $Bd(K)$ is a continuous curve.
Will it imply: $Bd(K)$ is homotopic to a point in $D$.
Question arise from: Understanding Rouche theorem statement given on wiki https://en.m.wikipedia.org/wiki/Rouché%27s_theorem
Proof uses Argument principle but argumen principle is true for curves homotopic to a point.
Question 2: As example given by M. Winter, this is not true if $Bd(K)$ is simply image of a continuous loop. What if it is image of 1-1 continuous loop i.e. homeomorphic to $S^1$ i.e. image of simple loop.
This is not true in general, depending on your exact definitions. Consider the following figure:
The (interior of the) lighter area is $D$, the (interior of the) darker area is $K$. The boundary of $K$ can be considered the embedding of a closed curve which winds around the annulus twice, henc cannot be contracted to a point in $D$.