now here's one of those things that confuse me to no end in topology. (i never took topology officially, and the text i have is like a crash course in topology, so it's not good to learn from). but anyway, somewhere in the text they said [0,1] was an open set
here's the context:
they were considering the subspace . The claim was, this subspace is not connected since we can express it as the union of two dis-joint, open sets, namely [0,1] and (2,3). Now instead of explaining to a dunce like me why [0,1] is open in the topology of X, the book simply warns, "remember, folks, we're dealing with a topology of X and not that of "
i tried looking back through the text to see what the author had said to make him believe this should not be a surprise to me...to no avail.
the best i could come up with is the definition, "Let be a topological space. The set is called closed when is open"
now i could not reconcile this with the claim [0,1] is open. is not , which is open, so should be closed?