Separation Axioms on Double Pointed Topology/T3.5 Axiom
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Theorem
Let $T_1 = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete topology on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_1 \times D, \tau}$ be the double pointed topological space on $T_1$.
Then $T \times D$ is a $T_{3 \frac 1 2}$ space if and only if $T$ is also a $T_{3 \frac 1 2}$ space.
Proof
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Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Notes: Part $1$: Basic Definitions: Section $2$. Separation Axioms: $1$