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The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.

The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object.

几何对象X的尺寸的直观概念是指需要多少个独立参数才能找到一个独特的点。但是，任何由两个参数指定的点都可以由一个参数指定，因为为实平面的基数等于实线的基数(这可以通过交织两个数字以产生一个编码相同信息的单个数字看到)。'''<font color = '#ff8000'>皮亚诺曲线</font>'''的例子表明，可以将实线完美和连续地映射到实平面(把一个实数转换成一对实数，从而覆盖所有实数对)，由此一维对象完全填充了一个高维对象。

几何对象X的尺寸的直观概念是指需要多少个独立参数才能找到一个独特的点。但是，任何由两个参数指定的点都可以由一个参数指定，因为为实平面的基数等于实线的基数(这可以通过交织两个数字以产生一个编码相同信息的单个数字看到)。'''<font color = '#ff8000'>皮亚诺曲线</font>'''的例子表明，可以将实线完美和连续地映射到实平面(把一个实数转换成一对实数，从而覆盖所有实数对)，由此一维对象完全填充了一个高维对象。