Topology: Question on why this is a closed set The excerpt is from "Topology Without Tears" by Sidney Morris. I am not able to get the rationale for the steps (1) and (2) in the above picture. Why does "[0,1] = [0,1]$$\cap$$Y " imply that [0,1] is closed in (Y,$$T_1$$) ? And why "[0,1]=(-1,1.5)$$\cap$$Y" imply [0,1] is open in (Y,$$T_1$$) ?

My reasoning for why [0,1] is clopen is as follows. Since (Y,$$T_1$$) = [0,1]$$\cup$$[2,3], by definition [0,1] is in this topology, hence it is open. Since it is the complement of [2,3] in Y, it is closed. Why taking an intersection with arbitrary sets is required as in the above picture? What have I misunderstood here?

• Do you know what the "subspace" topology is? Also, concerning the part : "since $(Y,T_1) = [0,1] \cup [2,3]$, by definition $[0,1]$ is in this topology, hence it is open" : note that $[0,1]$ is open if and only if it is contained in $T_1$, so how do you know if it is contained in $T_1$ or not? Finally, do you have a definition of $T_1$, or a characterization of the sets in $T_1$? If you do not, then this is what the subspace topology is all about,and you are missing some definitions required to complete the proof. – астон вілла олоф мэллбэрг Feb 5 at 14:52
• When he says $(Y,T_1)=[0,1]\cup[2,3],$ he means $Y=[0,1]\cup[2,3]$ with the relative topology. He isn't saying that the open sets in $T_1$ are $[0,1]$ and $[2,3]$. – saulspatz Feb 5 at 14:54

I don't understand your argument "Since $$(Y,T_1)=[0,1]\cup[2,3]$$ by definition $$[0,1]$$ is in this topology, hence it is open". What you need here is to know the definition of topological subspace. If $$(X,T)$$ is a topological space and $$A\subset X$$ then you can define a topology on $$A$$ like this: $$T_A=\{A\cap U: U\in T\}$$. And then $$(A,T_A)$$ is called a subspace of $$(X,T)$$. So in your case the open sets in $$Y$$ are the sets which can be written as an intersection of an open set in $$\mathbb{R}$$ (with the standard topology) with $$Y$$. Also it is easy to prove that the closed sets in $$Y$$ are the sets which can be written as an intersection of a closed set in $$\mathbb{R}$$ with $$Y$$.