Topologies on Set with More than One Element may not be Homeomorphic

Theorem

Let $S$ be a set which contains at least $2$ elements.

Let $\tau_1$ and $\tau_2$ be topologies on $S$.

Then it is not necessarily the case that $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ are homeomorphic.

Proof

Let $\tau_1$ be the indiscrete topology on $S$.

Let $\tau_2$ be the discrete topology on $S$.

Then $\struct {S, \tau_1}$ has $2$ elements: $S$ and $\O$.

Let $a, b \in S$ such that $a \ne b$.

Then $\set a \in \tau_2$ and $\set b \in \tau 2$, as well as $S$ and $\O$.

So there cannot be a bijection between $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$.

$\blacksquare$

Sources

• 1964: Steven A. Gaal: Point Set Topology ... (previous) ... (next): Chapter $\text {I}$: Topological Spaces: $1$. Open Sets and Closed Sets