Topology MOC

Metric topology

A metric topology is a topology formed on a Metric space , using open balls as a topological basis.

Given an arbitrary topological space , if there exists a metric that forms a metric space equivalent to , then the topological space is said to be metrizable.

Open and closed sets

For a metric space , a subset is called open iff for every point , there exists some such that the Open ball . In other words, is a Neighbourhood of every : one can move in any direction from a point without leaving .

As is always the case with a topological space, a set is called closed iff its compliment is open. See also Sequential closedness.

Loosely speaking, in the standard euclidean metric space , a set is open iff it does not include its boundary, while it is closed iff it does.

Outside of the trivial clopen sets and , clopen sets can occur in metric spaces when the boundary of a subset is not included in the space. For example, in the metric space with standard metric, the set is clopen since the boundary . A topology in which all sets are clopen is defined by the discrete metric.

Basic properties

Since a metric topology forms a topological space,

• the (in)finite union of open sets is open.
• the finite union of closed sets is closed.
• the finite intersection of open sets is open.
• the (in)finite intersection of closed sets is closed.

Properties

• Metrizable implies Hausdorff (see Hausdorff)
• Metrizable implies first-countable (see First countability axiom)

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