Why is the set not complete?

Good evening, while solving the problem "Is (1,3] a closed set?" I don't understand the step with r in the solution (see appendix) and what exactly is being shown here. I would be very grateful for any clarification on this. Best regards

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plankton986
3 months ago

I do not take the step with the r in the solution (see Annex) and what is shown here

This shows that there is no epsilon environment around the 1 that is completely incomplement.

plankton986
3 months ago

Because this creates a gap between the intervals that is not in the amount.