Pseudocompactness is continuous image-closed
This article gives the statement, and possibly proof, of a topological space property (i.e., pseudocompact space) satisfying a topological space metaproperty (i.e., continuous image-closed property of topological spaces)
View all topological space metaproperty satisfactions | View all topological space metaproperty dissatisfactions
Get more facts about pseudocompact space |Get facts that use property satisfaction of pseudocompact space | Get facts that use property satisfaction of pseudocompact space|Get more facts about continuous image-closed property of topological spaces
Given: A topological space such that, for any continuous map , is bounded. A continuous map .
To prove: For any continuous map , is bounded.
Proof: Note first that since is continuous, so is . Henceforth, we think of as a map from to .
Consider a continuous map . Define . Then, is a map. Since both and are continuous, so is , and by assumption, we therefore have bounded. But by definition, so is bounded, completing the proof.