Immersions are continuous

Question

I am new to topology, and I am wondering if this is right.
If $M, N$ are metric spaces, and $X\subset M$, $\forall x\in X$, $j_{x}:N\rightarrow \{x\}\times N$, with $j_{x}(y)=(x,y)$ is a continuous function because $d(j_{x}(y_{1}),j_{x}(y_{2}))=d((x,y_{1}),(x,y_{2}))=d(y_{1},y_{2})$, so for $d(y_{1},y_{2})<\delta$, and $\delta=\epsilon$, we have $d(j_{x}(y_{1}),j_{x}(y_{2}))<\epsilon$.
I am not sure if I could use that $d((x,y_{1}),(x,y_{2}))=d(y_{1},y_{2})$.

Answer 1

You can put several equivalent metrics on $M \times N$, e.g. the sum metric $d_s((x_1, y_1), (x_2, y_2))=d_M(x_1,x_2) + d_N(y_1,y_2)$ or the max metric $d_\infty((x_1, y_1), (x_2, y_2))=\max(d_M(x_1,x_2),d_N(y_1,y_2))$. Both (and others) induce the product topology on $M \times N$ and both have the property that $D(j_x(y_1), j_x(y_2)) = d_N(y_1, y_2)$ for $D \in \{d_s, d_\infty\}$

So you could justify continuity (isometry even) that way but be explicit in the choices of metrics. Your notation could in general be improved upon: $x \times N$ is non-standard, maybe better use $\{x\} \times N$ or maybe you just mean $X \times N$?