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This is more of a linguistic question than a mathematical one, but let me try anyway:

What is the antonym of subspace?

Is it a superspace? What else?

Suppose $X$ is a subspace of $Y$, say, a topological space, a vector space or whatever other spaces there are in mathematics. What is then $Y$ in relation to $X$?

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    @roman Would be be interested in posting some variation of your comment as an answer, so that this no longer remains on the unanswered page?2014-01-01

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If $X$ is a subspace of $Y$, then $Y$ is usually called the ambient space of $X$.