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Consider the $\mathbb{R}^3$ space with the standard Euclidean metric $\rho(p, q) = \sqrt{(p_x-q_x)^2 + (p_y - q_y)^2 + (p_z - q_z)^2}$. Consider the topology induced not only by open balls, but also by open discs (where disc with the center $a$ and radius $R$ is the set of points $D = \{x \mid \rho(x, a) < R\}$ such that entire $D$ lies on one plane) and open intervals (where interval with the center $a$ and radius $R$ is the set of points $I = \{x \mid \rho(x, a) < R\}$ such that entire $I$ lies on one line). Is this topology interesting in some way? Are there some funny properties? What reasonings about $R^3$ that are correct in usual metric topology will be broken in such a topology?

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    $\begingroup$ It looks like this way we get the discrete topology. Each set consisting of exactly one point is an intersection of two intervals, so each such set is open, so every set will be open. $\endgroup$ – Dan Shved Jan 19 '13 at 12:04
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In fact, your topology is discrete: the intersection of an open disk and a perpendicular line is reduced to a singleton.

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