Hey there, I don't know if anyone can help with this or not but what the heck.
Y=(1,5]. Which of the following subsets of Y are open and which are closed in Y, in the standard topology?
(0,1), (0,1], [1,2]
Thanks
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 $\displaystyle X = [0,1] \cup (2,3) \subset \mathbb{R}$. 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 $\displaystyle \mathbb{R}! \cdots$"
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 $\displaystyle X$ be a topological space. The set $\displaystyle A \subset X$ is called closed when $\displaystyle X \backslash A$ is open"
now i could not reconcile this with the claim [0,1] is open. is not $\displaystyle X \backslash [0,1] = (2,3)$, which is open, so $\displaystyle [0,1]$ should be closed?