Can We Show Y=X If Y is a Subspace of X and Y^c is First Category?”

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In the discussion, participants explore the conditions under which a subspace \( Y \) of a f-space \( X \) can be shown to equal \( X \) when the complement \( Y^{c} \) is of first category in \( X \). The consensus is that under these conditions, it is indeed possible to conclude that \( Y = X \). The conversation emphasizes the importance of including relevant definitions for clarity and suggests using LaTeX formatting for mathematical expressions to enhance readability.

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Suppose that $$X$$ is a f-space and $$Y$$ is a subspace of $$X$$ and $$Y^{c}$$ is a first category in $$X$$. Can we show $$Y=X$$?
 
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More people will be able to answer if you include the relevant definitions. Yes, we can look them up, but we shouldn't have to. The math in your post would have looked better if you had typed ## instead of $$. You can find the instructions on how to use LaTeX here by scrolling to the top, and clicking site info, frequently asked questions.
 

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