# How to read a chapter about connectedness for topological spaces as if you only want to know things about metric spaces?

I am reading chapter 12 of the book "Introduction to Metric&Topological Spaces" written by Wilson A. Sutherland, because I follow a course is about metric spaces. I missed some classes.

The author talks about connectedness of topological spaces. But I don't even know what topological spaces and for this course, I don't need to know what they are, we just focus on metric spaces because that should be less abstract/difficult to deal with.

So my main question is: how can I use definitions of connectedness and pathconnectedness for topological spaces as if they were metric spaces? I think I cannot just interchange the word 'topological' by 'metric'.

So for example, I would like to know how to deal with the following definitions, theorems and properties:

• A topological sapce $X$ is connected iff there does not exist a continuous map from $X$ onto a two-point discrete space (for example {$0,1$}$\subset\mathbb{R}$.

• A partition {$A,B$} of a topological space $X$ is a pair of non-empty subsets $A$ and $B$, such that $X=A\cup B$ and $A\cap B = \emptyset$, both are open in $X$

• Topological space $X$ is connected $\iff$ it admits no partition

• Topological space $X$ is connected $\iff$ the only subsets of $X$ which are both open and closed in $X$ are $X, \emptyset$

• A topological space $X$ is pathconnected if any two points in $X$ can be joined by a path in $X$ (where a path just means a continuous map $f:[0,1] \rightarrow X$ such that $f(0)=x, f(1)=y$

• Pathconnectedness implies connectedness in any topological space $X$

Any help is appreciated. If one can give me some online resources about connectedness of just metric spaces I would like to hear that as well :-)