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?