Some doubt regarding space filling curve.
Peano's idea was to define a sequence of curves that visit ever more points than the previous. Hence the limiting curve will visit a dense set of points in the square.
My doubt: we need to show that the limiting curve is onto, not to show it has dense range.
My thought: limiting curve will be continuous, continuous image of a compact set is compact, hence the range will be closed. A closed, dense set will be the whole space and hence the map will be onto.
Question: Why the limiting curve exists and continuous?
continuity compactness curves
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
add a comment |
Peano's idea was to define a sequence of curves that visit ever more points than the previous. Hence the limiting curve will visit a dense set of points in the square.
My doubt: we need to show that the limiting curve is onto, not to show it has dense range.
My thought: limiting curve will be continuous, continuous image of a compact set is compact, hence the range will be closed. A closed, dense set will be the whole space and hence the map will be onto.
Question: Why the limiting curve exists and continuous?
continuity compactness curves
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
2
That's true. So what's the question?
– Robert Israel
Dec 10 '18 at 4:22
add a comment |
Peano's idea was to define a sequence of curves that visit ever more points than the previous. Hence the limiting curve will visit a dense set of points in the square.
My doubt: we need to show that the limiting curve is onto, not to show it has dense range.
My thought: limiting curve will be continuous, continuous image of a compact set is compact, hence the range will be closed. A closed, dense set will be the whole space and hence the map will be onto.
Question: Why the limiting curve exists and continuous?
continuity compactness curves
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
Peano's idea was to define a sequence of curves that visit ever more points than the previous. Hence the limiting curve will visit a dense set of points in the square.
My doubt: we need to show that the limiting curve is onto, not to show it has dense range.
My thought: limiting curve will be continuous, continuous image of a compact set is compact, hence the range will be closed. A closed, dense set will be the whole space and hence the map will be onto.
Question: Why the limiting curve exists and continuous?
continuity compactness curves
continuity compactness curves
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
edited Dec 10 '18 at 6:39
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
asked Dec 10 '18 at 3:26
Santanu Debnath
1529
1529
2
That's true. So what's the question?
– Robert Israel
Dec 10 '18 at 4:22
add a comment |
2
That's true. So what's the question?
– Robert Israel
Dec 10 '18 at 4:22
2
2
That's true. So what's the question?
– Robert Israel
Dec 10 '18 at 4:22
That's true. So what's the question?
– Robert Israel
Dec 10 '18 at 4:22
add a comment |
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2
That's true. So what's the question?
– Robert Israel
Dec 10 '18 at 4:22