Rudin defines the segment to be the set of all real numbers such that .
In one example he considers the subset of and says that it is not an open set if we regard it as a subset of , but that it is open if we regard it as a subset of .
How do I regard the segment as a subset of ? The way I see it from his definition of a segment, it is just a part of the "x-axis" and would be open just as in ...
This is a pretty old thread, but I'm taking the liberty of raising an unresolved issue with the original question.
While an explanation for being non-open in has been suggested, it does not explain how may be regarded as a subset of , as Rudin implies in pg. 35 "Example 2.2(g) showed that a set may be open relative to Y without being an open subset of X." In short, how is the interval open and non-open in the same metric space?
in general, the way we regard as a subset of is to identify it with the set .
this sends the point (the real line becomes "the x-axis").
the non-null intersection of an open disk in with , is just an open interval on the x-axis.
these "open intervals on the x-axis" are open in the relative topology on the x-axis induced by the topology on the plane, but are not open in the full plane.