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# Is the inclusion always continuous?

From Mathematics Is A Science

Jump to navigationJump to searchQuestion: Is the inclusion $f:X\rightarrow Y$ of a subset $X$ into a topological space $Y$ always continuous?

Answer: Yes.

Why: For any neighborhood $V$ of $f(a)$, $V'=V\cap X$ is a neighborhood of $a$ in $X$. So we have $$f(V')\subset V.$$ See relative topology. $\blacksquare$