always room
The reason is that each number in the interval feels like it is out in the open - that there is room around it. It may be a very small surrounding space around it, but room none-the-less.
We extend this idea that there is always room “around” each number to open sets in .
Open Sets in
In an open set in , each number has room on either side.
The idea is that inside an open set, every number sits inside an open interval inside the set. There is space between each number and the endpoint. There is a positive distance between each number and each endpoint.
Forcing a positive distance between each number and the endpoints turns out to have
far reaching consequences, which we will investigate in this course. First, we need
some language for open sets.
Open intervals cannot contain their endpoints.
The empty set, , is the set with no element. The empty set is a subset of the real
numbers. It is the subset containing no real numbers.
is an open set.
These types of statements are often said to be vacuously true. They are true, because there is nothing that violates the definition - because there isn’t anything at all.
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more examples can be found by following this link
More Examples of Real-Valued Functions