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2
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Could one find a counterexample that a topology space X is Tychonoff, seperable but hasn't a $G_\delta$-diagonal? A topology space has a $G_\delta$-diagonal when there is a sequence ${G_n}$ of open sets belonging to $X^2$ with the diagonal $\Delta$ = $\cap{G_n}$. |
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7
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The product space $[0,1]^\kappa$ for $\aleph_1\le\kappa\le\mathfrak c$ is compact $T_2$ (hence Tychonoff) and separable (by the Hewitt–Marczewski–Pondiczery theorem), but it does not have a $G_\delta$ diagonal (in fact, if a compact $T_2$ space has a $G_\delta$ diagonal, then its unique uniform structure has a countable fundamental system, hence it is metrizable). |
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