Weak Countable Compactness is not Preserved under Continuous Maps
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Theorem
Let $T_A = \struct {S_A, \tau_A}$ be a topological space which is weakly countably compact.
Let $T_B = \struct {S_B, \tau_B}$ be another topological space.
Let $\phi: T_A \to T_B$ be a continuous mapping.
Then $T_B$ is not necessarily weakly countably compact.
Proof
Let $\Z_{>0}$ be the strictly positive integers:
- $\Z_{>0} = \set {1, 2, 3, \ldots}$
Let $T_A = \struct {\Z_{>0}, \tau_A}$ be the odd-even topology.
Let $T_B = \struct {\Z_{>0}, \tau_B}$ be the discrete topology on $\Z_{>0}$.
Let $\phi: T_A \to T_B$ be the mapping:
- $\map \phi {2 k} = k, \map \phi {2 k - 1} = k$
Then:
- $\map {\phi^{-1} } k = \set {2 k, 2 k - 1} \in \tau_A$
demonstrating that $\phi$ is continuous.
Now we have that the Odd-Even Topology is Weakly Countably Compact.
But we also have that a Countable Discrete Space is not Weakly Countably Compact.
Hence the result.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $6$. Odd-Even Topology: $4$