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Suppose that T is a topology on R that contains the set of all closed intervals. Prove that T is the discrete topology on R.

I am struggling to understand all of the different topologies. If anyone has an idea of where to do I would appreciate that.

My definition for a discrete topology simply says Let X be any set. The collection D of all subsets of X is a topology for X. This collection D is called the discrete topology.

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  • $\begingroup$ The wording of the question is a little confusing! I wish it had said "contains all intervals of the form $[a, b]$" instead of "contains all closed intervals," since you're not working with the standard topology on $\mathbb{R}$. $\endgroup$ – hunter Apr 12 '15 at 17:26
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HINT: For each $x\in\Bbb R$, $[x,x]=\{x\}$ is a closed interval. If your definition of interval does not allow such degenerate intervals, then note that $[x-1,x]\cap[x,x+1]=\{x\}$. Thus, $\{x\}$ is open for each $x\in\Bbb R$. Use this to show that $T$ is the discrete topology, i.e., that $T=\wp(\Bbb R)$.

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