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 :-)

Thank you guys in advance

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    $\begingroup$ Note a metric space is a special case of a topological space: the open balls make up a basis for the open sets. You can try to translate the above in terms of open balls. $\endgroup$ – Pedro Tamaroff Mar 13 '13 at 14:39
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    $\begingroup$ Every statement that you list is true of metric spaces without change. $\endgroup$ – Brian M. Scott Mar 13 '13 at 14:59
  • $\begingroup$ Brian M Scott: thanks for your reaction $\endgroup$ – Applied mathematician Mar 13 '13 at 15:04

You can in fact just replace the word topological by the word metric. However, when you do this, you are secretly replacing the words open and closed by their metric space equivalents - they have a more general meaning in topological spaces, but when your topological space is in fact a metric space, they mean what they usually do.

This may not apply to everything the book says (it may for example give some examples using topological spaces that aren't metrizable), but it does apply to all of the bullet points you have listed.


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