Definition:Coarser Topology
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Definition
Let $S$ be a set.
Let $\tau_1$ and $\tau_2$ be topologies on $S$.
Let $\tau_1 \subseteq \tau_2$.
Then $\tau_1$ is said to be coarser than $\tau_2$.
This can be expressed as:
- $\tau_1 \le \tau_2 := \tau_1 \subseteq \tau_2$
Strictly Coarser
Let $\tau_1 \subsetneq \tau_2$.
Then $\tau_1$ is said to be strictly coarser than $\tau_2$.
This can be expressed as:
- $\tau_1 < \tau_2 := \tau_1 \subsetneq \tau_2$
Also known as
The terms weaker or smaller are often encountered, meaning the same thing as coarser.
Unfortunately, the term stronger is also sometimes encountered, meaning exactly the same thing.
To remove any ambiguity as to which one is meant, it is recommended that coarser be used exclusively.
Also see
- Definition:Finer Topology, the opposite of coarser topology.
- Definition:Discrete Topology
- Definition:Indiscrete Topology
- Discrete Topology is Finest Topology
- Indiscrete Topology is Coarsest Topology
Sources
- 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $1$: General Introduction
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): coarser (of a topology)
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.3$: The normed space $\map {CL} {X, Y}$. Strong and weak operator topologies on $\map {CL} {X, Y}$