Re:Re:metric 관련 문제 입니다.

작성자비는아픔|작성시간11.07.24|조회수46 목록 댓글 1

쓰고나니 notation이 너무 조잡스럽네요...;

$d$ : a metric on $(0,1)\subset\mathbb{R}$ defined by $\displaystyle d(x,y)=|\frac{1}{x}-\frac{1}{y}|$

$d_{u}$ : the standard metric on $\mathbb{R}$

Suppose $D$ is an extended metric of $d$ on $\mathbb{R}$

and the induced topology $T_{D}$ by the metric $D$ is equal to the standard topology $T_{d_{u}}$ of $\mathbb{R}$ .

Let $(\mathbb{R}^{2},T_{P_{D}})$ be the product topology of $(\mathbb{R},T_{D})$

and $(\mathbb{R}^{2}, T_{P_{d_{u}}})$ be the product topology of $(\mathbb{R},T_{d_{u}})$ .

Observe that $\displaystyle D:(\mathbb{R}^{2},T_{P_{D}})\rightarrow(\mathbb{R}_{\cup\{0\}}^{+},d_{u}|_{\mathbb{R}_{\cup\{0\}}^{+}})$ is continuous.(well-known)

Consider a sequence $(a_{n})_{n\in\mathbb{N}}$ on $\mathbb{R}^{2}$ with $\displaystyle a_{n}=(\frac{1}{n+1},\frac{1}{n+2})$ .

Then $\displaystyle \lim_{n\rightarrow\infty} D(a_{n}) = D(\lim_{n\rightarrow\infty} a_{n}) = D(0,0) = 0$ since $(a_{n})_{n\in\mathbb{N}}$ converges to $(0,0)$ over $T_{P_{D}}=T_{P_{d_{u}}}$ .