Does connectedness of a subspace $A$ depend on what space $A$ is a subspace of?

Question

Let $X$ be a topological space and let $Y$ be a subspace of $X$. If $A$ is a connected subspace of $X$ and if $A\subset Y$, then is $A$ a connected subspace of $Y$? Also, if $A$ is a connected subspace of $Y$ and if $Y\subset X$, then $A$ is a connected subspace of $X$?

In other words, does connectedness of a subspace A depend on what space A is a subspace of?

Thank you.

Answer 1

Like compactness, connectedness is absolute in the sense that it does not matter in what space it is embedded. Only the (subspace) topology matters.