Just a quick question - is a straight line that goes on indefinitely viewed as a closed set, open set or neither? Seeing as it includes all the boundary points as it travels, but it doesn't have any end points.

(E.g. $y = 2x$)

Thanks in advance!

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    $\begingroup$ Note that a set isn't naturally "closed" or "open" on its own: these are not properties that are intrinsic to a set. Rather, being open or closed depends on the topology of the ambient space. For instance, the line isn't open in the Euclidean topology of the plane, but it is open in itself. $\endgroup$ – Alex Provost Jan 25 '16 at 16:36
  • $\begingroup$ Note that "open" and "closed" are not mutually exclusive. However, since the plane is connected, the only sets that are both open and closed are the entire plane $\Bbb R^2$ and the empty set $\emptyset$. $\endgroup$ – Akiva Weinberger Jan 25 '16 at 17:14

A line is a closed subset of the plane -- as you have already seen by applying the definition. The definition does not say anything about "end points" or lack of same.

It is not an open subset of the plane, because an open subset has to contain a disc centered on every point in it, but the line contains no discs at all.

  • 1
    $\begingroup$ Thank you so much for your answer, it was really helpful! $\endgroup$ – Wildawg Jan 25 '16 at 16:44

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