Topological space

Coarseness and fineness of topologies

Given two topologies on the same set $(𝑋, T)$ , $(𝑋, T')$ , if $T \subseteq T'$ then $T$ is said to be coarser¹ than $T'$ , since it contains larger chunks of $𝑋$ in a smaller quantity. Likewise $T'$ is finer² than $T$ . Clearly all topologies are coarser than the Discrete topology and finer than the Trivial topology.³ #m/def/topology

$T 1 \subseteq T 2$ ::: $T 1$ is coarser than $T 2$
$T 1 \supseteq T 2$ ::: $T 1$ is finer than $T 2$

Properties

An intersection of topologies will clearly be coarser than both topologies.
Given two topologies $T 1, T 2$ on the same set $𝑋$ , if $𝑒 : T 1 \to T 2 : 𝑥 \mapsto 𝑥$ is continuous, then clearly $T 1 \supseteq T 2$ , i.e. $T 1$ is finer.

#state/tidy | #SemBr | #lang/en | #flashcards/maths/topology

German gröber ↩
German feiner ↩
2020, Topology: A categorical approach, §0.1, p. 2 ↩