Metric topology

A metric topology is a topology formed on a Metric space $(𝑀, 𝑑)$ , using open balls as a topological basis.

Given an arbitrary topological space $(𝑋, T)$ , if there exists a metric $𝑑 : 𝑋 \times 𝑋 \to ℝ$ that forms a metric space equivalent to $T$ , then the topological space $𝑋$ is said to be metrizable.

Open and closed sets

For a metric space $(𝑀, 𝑑)$ , a subset $𝑆 \subseteq 𝑀$ is called open iff for every point $𝑠 \in 𝑆$ , there exists some $𝛿 > 0$ such that the Open ball $𝐵 𝛿 (𝑠) \in 𝑆$ . In other words, $𝑆$ is a Neighbourhood of every $𝑠 \in 𝑆$ : one can move in any direction from a point $𝑠 \in 𝑆$ without leaving $𝑆$ .

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 $(ℝ 3, 𝑑)$ , a set is open iff it does not include its boundary, while it is closed iff it does.

Outside of the trivial clopen sets $𝑋$ and $\emptyset$ , 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 $(ℚ, 𝑑)$ with standard metric, the set ${𝑥 \in ℚ ∣ 𝑥 2 < 3}$ is clopen since the boundary $\pm \sqrt 3 \notin ℚ$ . 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

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