Topology MOC

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 ×𝑋 →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 ⊆𝑀 is called open iff for every point 𝑠 ∈𝑆, there exists some 𝛿 >0 such that the Open ball 𝐵𝛿(𝑠) ∈𝑆. In other words, 𝑆 is a Neighbourhood of every 𝑠 ∈𝑆: one can move in any direction from a point 𝑠 ∈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 (ℝ3,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 ∅, 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 ∈ℚ ∣𝑥2 <3} is clopen since the boundary ±√3 ∉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

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

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