Give an example of a metric space which admits an isometry with a proper subset itself. (Hint: Try Example 4)
Answer (1)
A metric space is a set where a distance (called a metric) is defined between elements of the set. An isometry in a metric space is a distance-preserving map from the space to itself.
To provide an example, consider the metric space ( R , d ) , where R is the set of all real numbers and d is the standard Euclidean distance. The function f : R → R defined by f ( x ) = 2 x is not an isometry because it does not preserve distance. However, the reflection map g : R → R defined by g ( x ) = − x is an isometry because:
It preserves distance, since for any two real numbers x , y , the distance between them remains unchanged: d ( g ( x ) , g ( y )) = ∣ g ( x ) − g ( y ) ∣ = ∣ − x − ( − y ) ∣ = ∣ x − y ∣ = d ( x , y ) .
Consider the proper subset ( 0 , ∞ ) of R . The map g takes elements of ( 0 , ∞ ) into the subset ( − f , 0 ) , which is a proper subset of R .
Therefore, the reflection map g ( x ) = − x on the real numbers R serves as an example of an isometry from a metric space into a proper subset of itself.
