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djoke
djoke
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Assume that $(X,\tau)$ is a topological space and assume that every continuous mapping $f$ of $X$ into real line $\mathbb{R}$ achieves its maximum. Under which conditions on $\tau$, the space $X$ is compact. It can be easily prove that, $X$ is compact, provided that $\tau$ is a metric topology in $X$.

Is for example this true for the Hausdorff spaces?

Assume that $(X,\tau)$ is a topological space and assume that every continuous mapping $f$ of $X$ into real line $\mathbb{R}$ achieves its maximum. Under which conditions on $\tau$, the space $X$ is compact. It can be easily prove that, $X$ is compact, provided that $\tau$ is a metric topology in $X$.

Assume that $(X,\tau)$ is a topological space and assume that every continuous mapping $f$ of $X$ into real line $\mathbb{R}$ achieves its maximum. Under which conditions on $\tau$, the space $X$ is compact. It can be easily prove that, $X$ is compact, provided that $\tau$ is a metric topology in $X$.

Is for example this true for the Hausdorff spaces?

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djoke
djoke
  • 303
  • 1
  • 8

Counterpart of Weierstrass theorem

Assume that $(X,\tau)$ is a topological space and assume that every continuous mapping $f$ of $X$ into real line $\mathbb{R}$ achieves its maximum. Under which conditions on $\tau$, the space $X$ is compact. It can be easily prove that, $X$ is compact, provided that $\tau$ is a metric topology in $X$.