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Keywords:
ring of continuous functions; $C^*$-embedded; $P$-point
ring of continuous functions; $C^*$-embedded; $P$-point
Summary:
In a Tychonoff space $X$, the point $p\in X$ is called a $C^*$-point if every real-valued continuous function on $C\smallsetminus \{p\}$ can be extended continuously to $p$. Every point in an extremally disconnected space is a $C^*$-point. A classic example is the space ${\bf W}^*=\omega_1+1$ consisting of the countable ordinals together with $\omega_1$. The point $\omega_1$ is known to be a $C^*$-point as well as a $P$-point. We supply a characterization of $C^*$-points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space is a $C^*$-point. This process leads to many interesting new discoveries.
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