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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?

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    $\begingroup$ 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. $\endgroup$ – астон вілла олоф мэллбэрг Feb 5 at 14:52
  • $\begingroup$ 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 $[2,3]$. $\endgroup$ – saulspatz Feb 5 at 14:54
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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 $Y$.

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