L.S.H said:Let ##\mathbb{X}## be the set of all sequences in ##\mathbb{R}## that converge to ##0##. For any sequences ##\{x_n\},\{y_n\}\in\mathbb{X}##, define the metric ##d(\{x_n\},\{y_n\})=\sup_{n}{|x_n−y_n|}##. Show the metric space ##(\mathbb{X},d)## is separable. I understand that I perhaps need to find a countable dense set in ##\mathbb{X}##.
A metric space is a mathematical structure consisting of a set of elements and a distance function that assigns a non-negative value to the distance between any two elements in the set. The distance function must satisfy certain properties, including the non-negativity, symmetry, and triangle inequality properties.
A metric space is said to be separable if it contains a countable dense subset. This means that there exists a subset of elements in the space that are "close" to every other element in the space, and this subset is countable, meaning it can be put into a one-to-one correspondence with the natural numbers.
Separability is an important property in metric spaces because it allows for the existence of a dense sequence of points. This can aid in the understanding and analysis of the space, as well as provide a useful tool for proving certain theorems.
To prove that a defined metric space is separable, one must show the existence of a countable dense subset in the space. This can be done by constructing such a subset or by showing that a known set is a dense subset of the space.
No, not all metric spaces are separable. In fact, there are many examples of metric spaces that are not separable, such as uncountable spaces or spaces with a "wild" topology. However, many commonly studied metric spaces, such as Euclidean spaces, are separable.