Homeomorphism may Exist between Non-Comparable Topologies
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Theorem
Let $S$ be a set.
Let $T_1 = \struct {S, \tau_1}$ and $T_2 = \struct {S, \tau_2}$ be topological spaces defined on the underlying set $S$.
Let $\tau_1$ and $\tau_2$ be non-comparable.
Then it may possibly be the case that $T_1$ and $T_2$ are homeomorphic.
Proof
A counterexample is demonstrated in Homeomorphic Non-Comparable Particular Point Topologies.
$\blacksquare$
Sources
- 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: Functions