Subspace Topology - Examples

Examples

In the following, R represents the real numbers with their usual topology.

  • The subspace topology of the natural numbers, as a subspace of R, is the discrete topology.
  • The rational numbers Q considered as a subspace of R do not have the discrete topology (the point 0 for example is not an open set in Q). If a and b are rational, then the intervals (a, b) and are respectively open and closed, but if a and b are irrational, then the set of all x with is both open and closed.
  • The set as a subspace of R is both open and closed, whereas as a subset of R it is only closed.
  • As a subspace of R, is composed of two disjoint open subsets (which happen also to be closed), and is therefore a disconnected space.
  • Let S = [0,1) be a subspace of the real line R. Then [0,½) is open in S but not in R. Likewise [½, 1) is closed in S but not in R. S is both open and closed as a subset of itself but not as a subset of R.

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