1
$\begingroup$

I am studying connectedness of topological spaces on my own, and I want to know if I got it right. Every connected space is not path connected. It happens so because we defined connectedness of two sets as intersection of a set and closure of other is not empty. This definition makes "every connected space is not path connected". Had the definition of connected been, sets who intersection is not empty, then every connected space would have been path connected. Am I correct?

Edit: it seems my question was bit unclear, so here is an example. In infinite deleted broom set, point (1,0) also belongs to the set. Now the intersection of limit points all those lines and (1,0) is not empty since (1,0) belongs to the intersection. Thus the deleted infinite broom is connected. And I know why it is not path connected, since there is no path from it to other lines, within the space. Thus every connected space is not path connected. I know about this fact and examples for it. Here is my question. I believe that every connected space is not path connected since I defined connectedness as non empty intersection of closure of one open set and other open set. But if I change the definition as " a space is disconnected if it can be written as union of two sets, which are disjoint" then every connected space will be path connected. Am I right?

$\endgroup$
  • $\begingroup$ What is connectedness of two sets? Connected property applies to one set, and the property is that it cannot be written as the disjoint union of two open sets. It is true that not every connected space is path connected, but the first counterexample one normally encounters is not the most obvious or immediate one. You have to think harder to find a counterexample, even with the ordinary definition. If a little tired, see the "topologist's sine curve". $\endgroup$ – астон вілла олоф мэллбэрг Nov 8 '16 at 9:03
  • 1
    $\begingroup$ no, the unit disk with the standard topology is both connected and path connected. Even more is true as path connected implies connected. The correct statement you are looking for is: In general connected does not imply path connected. $\endgroup$ – noctusraid Nov 8 '16 at 9:04
  • $\begingroup$ @noctusraid or perhaps "There are connected spaces which are not path connected" $\endgroup$ – Henry Nov 8 '16 at 9:07
  • $\begingroup$ @Henry That's another way to put it, yes. $\endgroup$ – noctusraid Nov 8 '16 at 14:21
  • $\begingroup$ One could also mention that the both are equivalent when we restrict ourself to (topological) manifolds only, which is pretty surprising (at least to me). $\endgroup$ – noctusraid Nov 8 '16 at 14:23
3
$\begingroup$

Your question is a bit unclear (you can try to clarify what you mean exactly by using more standard notation). In any case, the condition you seem to have in mind will not imply that a connected space is path-connected. To better understand the reltion between connectedness and path-connectedness, it is good to first understand that the definition of path-connectedness is a positive one in the sense that a space is path-connected if any two points in it can be connected by a path. Now, the definition of connectedness is a negative definition in the sense that a space is connected if it cannot be disconnected. That makes comparing the definitions a bit confusing. It may help to re-cast the definiton of connectedness in a positive fashion, where every two points are connected by a suitable something. This is done in metric characterization of connectedness for all topological spaces (top. Appl.)

$\endgroup$
  • $\begingroup$ I edited the question $\endgroup$ – jnyan Nov 8 '16 at 10:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.