Separation Properties Preserved by Expansion
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Theorem
These separation properties are preserved under expansion:
Proof
Let $S$ be a set.
Let $\struct {S, \tau_1}$ and $\struct {S, \tau_2}$ be topological spaces based on $S$ such that $\tau_2$ is an expansion of $\tau_1$.
That is, let $\tau_1$ and $\tau_2$ be topologies on $S$ such that $\tau_1 \subseteq \tau_2$.
Let $I_S: \struct {S, \tau_1} \to \struct {S, \tau_2}$ be the identity mapping from $\struct {S, \tau_1}$ to $\struct {S, \tau_2}$.
From Identity Mapping to Expansion is Closed, we have that $I_S$ is closed.
We also have Identity Mapping is Bijection.
So we can directly apply:
and hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text I$: Basic Definitions: Section $2$: Separation Axioms: Functions, Products, and Subspaces