- #1
Tio Barnabe
What are the ways of mathematically defining a space?
We usually define a space by stating some common property shared by the points that make up the space in question, that is, a symmetry of that space. For instance, the ordinary 2-sphere is defined as all points ##X## that are at a distance ##c > 0, c \in \mathbb{R}## from some point ##X_0##, the distance being calculated using some specific metric. Does this implicitaly define a coordinate system? It seems that it doesn't, as long as we take the usual definition of a coordinate system as the bijection from a region of the sphere to some other appropriate space through some specific mapping. (This was my first question.)
Now, say we have a real, arbritary object of every-day life; this could be my smartphone. Is there a similar way as above for defining it mathematically? I suppose that if there is, it must be extremely difficult, because the smartphone is a space (or object?) with no patterns... or does it has patterns, but these are so complex that a human brain could not realize them?
We usually define a space by stating some common property shared by the points that make up the space in question, that is, a symmetry of that space. For instance, the ordinary 2-sphere is defined as all points ##X## that are at a distance ##c > 0, c \in \mathbb{R}## from some point ##X_0##, the distance being calculated using some specific metric. Does this implicitaly define a coordinate system? It seems that it doesn't, as long as we take the usual definition of a coordinate system as the bijection from a region of the sphere to some other appropriate space through some specific mapping. (This was my first question.)
Now, say we have a real, arbritary object of every-day life; this could be my smartphone. Is there a similar way as above for defining it mathematically? I suppose that if there is, it must be extremely difficult, because the smartphone is a space (or object?) with no patterns... or does it has patterns, but these are so complex that a human brain could not realize them?
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