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

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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.
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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