Discrete Topology is Finest Topology

Theorem

Let $T = \struct {S, \tau}$ be a discrete topological space.

$\tau$ is the finest topology on $S$.

Let $\phi$ be any topology on $S$.

Let $U \in \phi$.

Then, by the definition of topology, $U \subseteq S$.

Then, by the definition of discrete topological space, $U \in \tau$.

Hence by definition of subset, $\phi \subseteq \tau$.

Hence by definition of finer topology, $\tau$ is finer than $\phi$.

$\blacksquare$

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
1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $1$
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 $\map {CL} {X, Y}$