Search
- Filter Results
- Location
- Classification
- Include attachments
- https://math.libretexts.org/Bookshelves/Analysis/Mathematical_Analysis_(Zakon)/03%3A_Vector_Spaces_and_Metric_Spaces/3.07%3A_Metric_SpacesThe distances \(\rho(x, y)\) in \(S\) are, of course, also defined for points of \(A\) (since \(A \subseteq S\), and the metric laws remain valid in \(A .\) Thus \(A\) is likewise a (smaller) metric s...The distances \(\rho(x, y)\) in \(S\) are, of course, also defined for points of \(A\) (since \(A \subseteq S\), and the metric laws remain valid in \(A .\) Thus \(A\) is likewise a (smaller) metric space under the metric \(\rho\) "inherited" from \(S ;\) we only have to restrict the domain of \(\rho\) to \(A \times A\) (pairs of points from \(A ) .\) The set \(A\) with this metric is called a subspace of \(S .\) We shall denote it by \((A, \rho),\) using the same letter \(\rho\) or simply by \…
- https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_junalyn2020/Book%3A_Introduction_to_Real_Analysis_(Lebl)/7%3A_Metric_Spaces/7.1%3A_Metric_SpacesNote that \(d(x,y) = \varphi(\left\lvert {x-y} \right\rvert)\) where \(\varphi(t) = \frac{t}{t+1}\) and note that \(\varphi\) is an increasing function (positive derivative) hence \[\begin{split} d(x,...Note that \(d(x,y) = \varphi(\left\lvert {x-y} \right\rvert)\) where \(\varphi(t) = \frac{t}{t+1}\) and note that \(\varphi\) is an increasing function (positive derivative) hence \[\begin{split} d(x,z) & = \varphi(\left\lvert {x-z} \right\rvert) = \varphi(\left\lvert {x-y+y-z} \right\rvert) \leq \varphi(\left\lvert {x-y} \right\rvert+\left\lvert {y-z} \right\rvert) \\ & = \frac{\left\lvert {x-y} \right\rvert+\left\lvert {y-z} \right\rvert}{\left\lvert {x-y} \right\rvert+\left\lvert {y-z} \righ…
- https://math.libretexts.org/Under_Construction/Purgatory/Remixer_University/Username%3A_junalyn2020/Book%3A_Introduction_to_Real_Analysis_(Lebl)/7%3A_Metric_SpacesIn mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces top...In mathematics, a metric space is a set for which distances between all members of the set are defined. Those distances, taken together, are called a metric on the set. A metric on a space induces topological properties like open and closed sets, which lead to the study of more abstract topological spaces.
