Adding a continuous map by forcing
Keywords:
Forcing, continuous map, homeomorphismAbstract
We study how forcing adds continuous maps. We prove the following theorems. Let $X$ be a completely regular space.
(1) If $X$ is a scattered compact Hausdorff space and $Y$ is a discrete space, then forcing does not add any continuous maps from $X$ to $Y$.
(2) If $X$ is not a zero-dimensional scattered pseudocompact space and $Y$ is a topological space which has more than one point, then some ccc forcing adds a continuous map from $X$ to $Y$.
(3) If $X$ is infinite and there is a homeomorphism from $X$ onto $X$ which is not the identity map, then some ccc forcing adds a homeomorphism from $X$ onto $X$.
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