Remark

Let Y be a topological space, and let X, $X'$ and $Y'$ be subspaces of Y such that $X\subset X'\subset Y'\subset Y$. Suppose that the canonical injection of X into Y is relatively connected. For every connected open subset V of Y, the set $V\cap X$ is connected and dense in V (IV, p. 354, cor. 2); consequently, the set $V\cap X'$ is connected (TG, I, p. 81, prop. 1). This proves that the canonical injection of $X'$ into $Y'$ is relatively connected.

Let V be an open subset of Y; by what precedes, the canonical injection of $V\cap X$ into $V\cap X'$ is relatively connected. If the set $V\cap X$ is simply connected, $V\cap X'$ is so also, by virtue of corollary 2 of IV, p. 357. Consequently, if the canonical injection of X into Y satisfies the hypotheses of proposition 6, the same is true of the canonical injection of $X'$ into $Y'$.