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closure and interior reference flip-flop
(reminder regarding topological terminology: boundary = closure minus interior)
Ordinary language flip-flops between the interior and the closure, depending on context to make it clear which is meant. For example, for a woman in the store with her small child, we might say that the child is either being carried by the woman in her arms, or is walking beside the woman, or is in the basket. If the child is sitting in the seating portion of the basket, we would normally say that the child is in the basket. However, the basket has a warning label saying “Do NOT allow child to ride in basket”. What the warning label is referring to, of course, is the large receptacle portion of the basket, in other words, the interior of the concept ‘basket’. The seating portion of the basket is the boundary of the basket. Another example is whether, in a court case, the underlying facts would support a felony conviction. This is the interior of the case. The ultimate disposition of the case – perhaps involving plea bargaining – is the closure (no pun intended) of the case. For another example, the voice-over of an online training program may tell you that you have completed the lesson – but that now you must take – and pass – a quiz on the lesson. Going through the material is the interior of the lesson. Passing the quiz afterwards is the boundary of the lesson. The combination of the two is the closure of the lesson. Another example is interval reference. Whether a certain one of the two endpoints is included in the reference can be a key issue, illustrated in the saying, “It’s not the fall that kills you.”
The interior is referenced in ordinary language via the postpositive adjective ‘proper’ – e.g., the child is not to ride in the basket proper. (A well-known example of a postpositive adjective is ‘Attorney General’. Another one is ‘sergeant-at-arms’.)
keywords: topology