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[tex]\forall A:x\quad \exists B:y[/tex]

Am I supposed to just leave a space before the [itex]\exists[/itex] or is there a symbol I should use?

Now consider the definition of a compact set. A subset K of a topological space X is said to be compact if every open cover of K has a finite subcover. How can I write this with logical symbols? Suppose I start by saying that K is a subset of a topological space with topology [itex]\tau[/itex], and then start the next sentence with "K is said to be compact if". How should I finish it? Here's a suggestion (which probably needs to be improved):

[tex]\{U_i\in\tau|i\in I\}: \bigcup_{i\in I}U_i\supset K\implies \exists I_0\subset I:\bigg(|I_0|<\infty\quad \bigcup_{i\in I_0}U_i\supset K\bigg)[/tex]

Can I use parentheses like that? Do I have to include an "and" symbol between the two statements in parentheses? Should I have made it a "for all" statement instead of making the whole thing an implication?

The next thing I'm going to ask about is the proper way to rewrite the statement in an equivalent way. For example, if we've written down an implication [itex]A\implies B[/itex], I'd like to rewrite it as [itex]\lnot B\implies \lnot A[/itex]. And if we've written down a "for all" statement, I'd like to rewrite that in a way that corresponds to what I just said I want to do to the implication.