I am an undergraduate students and there are some competitive exams like TIFR-GS,CMI,ISI-MMath,NBHM at the end of our undergraduate course.I have seen some problems on homeomorphism that appear to be of the same type but we are not introduced to solving such problems as they are not in our undergraduate course of metric spaces.Some of the problems are:
$1.$ Does there exists a continuous surjection from $\Bbb R^3-S^2$ to $\mathbb R^2-\{(0,0)\}$.
$2.$ Does there exist a continuous onto function from $S^2$ to $S^1$?
$3.$ There does not exists a continuous surjection from $S^1$ to $\mathbb R$.
$4.$ Show that $\mathbb R^n$ and $\mathbb R^m$ are not homeomorphic if $m\neq n$.
$5.$ Does there exist a homeomorphism between $S^n-\{0\}$ and $\mathbb R^n$ ?
These are somewhat like application of topological ideas in geometry but these types of problems are not covered in out course.
So,I have to study it on my own to get prepared for the mentioned exams.One of my professors suggested the book Basic Topology- M. A. Armstrong but unfortunately it is too advanced and there are so many things that I do not require at present,for example they use fundamental groups,homotopy,quotient maps etc.So,this book does not suit me or is too difficult for me.
So,can anyone please suggest me some reference book where I can learn how to approach such problems and is easy to interpret?