Then a subspace of is open in if and only if it is open in. In other words the subspace topology that inherits from is the same as the one it inherits from. If is a subspace of then is also a subspace of with the same topology.The closed sets in are precisely the intersections of with closed sets in.If is continuous the restriction to is continuous.We list some further properties of the subspace topology. This property is characteristic in the sense that it can be used to define the subspace topology on. Then for any topological space a map is continuous if and only if the composite map is continuous. Let be a subspace of and let be the inclusion map. The subspace topology has the following characteristic property. S is both open and closed as a subset of itself but not as a subset of R. Likewise [½, 1) is closed in S but not in R. Let S = [0,1) be a subspace of the real line R.As a subspace of R, is composed of two disjoint open subsets (which happen also to be closed), and is therefore a disconnected space.The set as a subspace of R is both open and closed, whereas as a subset of R it is only closed.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 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).The subspace topology of the natural numbers, as a subspace of R, is the discrete topology.In the following, R represents the real numbers with their usual topology. is then homeomorphic to its image in (also with the subspace topology) and is called a topological embedding. The open sets in this topology are precisely the ones of the form for open in.
Then the subspace topology on is defined as the coarsest topology for which is continuous. More generally, suppose is an injection from a set to a topological space. Subsets of topological spaces are usually assumed to be equipped with the subspace topology unless otherwise stated.Īlternatively we can define the subspace topology for a subset of as the coarsest topology for which the inclusion map If is equipped with the subspace topology then it is a topological space in its own right, and is called a subspace of.
That is, a subset of is open in the subspace topology if and only if it is the intersection of with an open set in. Given a topological space and a subset of, the subspace topology on is defined by 4 Preservation of topological properties.
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