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We now consider some ways of getting new topologies from old ones.

**Definition**

If *A* is a subset of a topological space (*X*, _{X}), we define the **subspace topology** _{A} on *A* by:

*B* _{A} if *B* = *A* *C* for some *C* _{X} .

**Examples**

- Restricting the metric on a metric space to a subset gives this topology.

For example, On*X*= [0, 1] with the usual metric inherited from**R**, the open sets are the intersection of [0, 1] with open sets of**R**.

So, for instance, [1,^{1}/_{4}) = (-1,^{1}/_{4}) [0, 1] and so is an open subset of the*subspace**X*.**Remark**

Note that as in this example, sets which are open in the subspace are not necessarily open in the "big space". - The subspace topology on
**Z****R**(with its usual topology/metric) is the discrete topology. - The subspace topology on the
*x*-axis as a subset of**R**^{2}(with its usual topology) is the usual topology on**R**.

If we take the inclusion map

If

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