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I am studying Topological spaces and specially metric spaces. Suppose [0,1] U [2,3] is a metric space with some predefined metric. The the question is, Is [0,1] is open or closed or both open and closed or neither open nor closed ?

My answer:

A) Since every point in an open set is an interior point, then [0,1] on the metric space is not open, since one can always construct a neighbourhood arround the point 1 or 0, whose intersection with the set consists of infinite points which are not in the set [0,1]. It seems it is closed set because it contains all of its accumulation points. Now if [0,1] is closed then so is [2,3]. But the complement of the closed set in a metric space must be open and therefore [2,3] is also open. This means [2,3] is both open and closed. Tha same is true for [0,1]. Therefore it is also both open and closed.

B) Every point in the set [0,1] is an interior point because the neighbourhood arround point 1(no matter how small it may be) consists of points which lies in the interval [0,1][tex]\bigcup[/tex][2,3]. It is because the set {x:[tex]\rho[/tex](x) [tex]\geq[/tex]1 } does not lie within the space. That means [0,1] is an open set, then same as [2,3]. but again the complement of an open set must be closed. so [2,3] must be closed. Thus [0,1] is both open and closed.

So out of this two explanation which one is correct. If explanation two is correct then does it mean that when we talk about intersection of interior points of a point a and the set containing a, the intersection set has to lie within the inherent metric space ? Does that mean set of natural numbers {1,2,3,4,...} on R is closed while the set {2,3} is open in the space {1,2,3,4,..}.

Please help me to sort this confusion. I am very novice in this field so please disregard if i made a very silly question.

regards

Titas

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# A Confusion about open sets

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