Discrete Topology is Topology

Theorem

Let $S$ be a set.

Let $\tau$ be the discrete topology on $S$.

$\tau$ is a topology on $S$.


Proof

Let $T = \struct {S, \tau}$ be the discrete space on $S$.

Then by definition $\tau = \powerset S$, that is, is the power set of $S$.

We confirm the criteria for $T$ to be a topology:

$(1): \quad$ By definition of power set, $\O \in \powerset S$ and $S \in \powerset S$.
$(2): \quad$ From Power Set with Union is Monoid, $\powerset S$ is closed under set union.
$(3): \quad$ From Power Set with Intersection is Monoid, $\powerset S$ is closed under set intersection.

$\blacksquare$


Sources

  • 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $3$: Continuity generalized: topological spaces: $3.1$: Topological Spaces: Example $3.1.4$
  • 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
  • 2011: John M. Lee: Introduction to Topological Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Topological Spaces: Topologies
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