Let the space $X=Y\cup\{a\}$, where $Y$ is a metriable subspace and $a \in \overline Y$. As we know, if $f$ is the coutinuous function from $Y^2$ to $I$, then $f$ can be continuously extendable over $X^2$. Since $Y$ is a metrizable space, there exist some continuous functions from $Y^2$ to $I$ satisfying $f(y_1,y_2)\neq 0$ whenever $y_1 \neq y_2$; $f(y_1,y_2)=0$ whenever $y_1 = y_2$. Therefore, the extensions $f^\sim (a,a)=0$. However, I don't know whether there exists one of those extensions satisfying that $f^\sim (a,y)\neq0$ and $f^\sim (y,a)\neq0$ for any $y \in Y$. Could someone help me?
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Let (C) denote the condition that a function $f$ defined on $Y\times Y$ is such that $f(x,y)\ne0$ for every $x\ne y$ in $Y$, $f(y,y)=0$ for every $y$ in $Y$. Here is a positive example: assume that $X$ is metrizable, let $g(x,y)=d(x,y)$ for every $x$ and $y$ in $X$, and let $f$ denote the restriction of $g$ to $Y\times Y$. Then $f^\sim=g$ is the unique continuous extension of $f$ to $X\times X$, $f$ fulfills condition (C) and $f^{\sim}(a,y)=f^{\sim}(y,a)\ne0$ for every $y$ in $Y$. Here is a negative example: assume that $X$ is metrizable, let $g(x,y)=d(x,y)d(a,x)d(a,y)$ for every $x$ and $y$ in $X$, and let $f$ denote the restriction of $g$ to $Y\times Y$. Then $f^\sim=g$ is the unique continuous extension of $f$ to $X\times X$, $f$ fulfills condition (C) and $f^{\sim}(a,y)=f^{\sim}(y,a)=0$ for every $y$ in $Y$. These two examples show that some continuous functions $f$ on $Y\times Y$ such that condition (C) holds have a continuous extension $f^\sim$ to $X\times X$ such that $f^{\sim}(a,y)=f^{\sim}(y,a)\ne0$ for every $y$ in $Y$ and that some others have not. |
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For all $y\in Y$, you will define $f^\sim (a,y) = \lim_{n \rightarrow\infty} f(y_n, y)$ for any sequence $y_n \mathop\longrightarrow\limits_{n\rightarrow\infty} a$ in Y. As $f$ is the distance on Y, if $\lim_{n \rightarrow\infty} f(y_n, y) = 0$ then $y_n \mathop\longrightarrow\limits_{n\rightarrow\infty} y$. By uniqueness of the limit, then $y = a$ and $a \in Y$, a contradiction. |
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