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Erschienen in:
2015 | OriginalPaper | Buchkapitel
A Taste of Topology
verfasst von : Charles C. Pugh
Erschienen in: Real Mathematical Analysis
Verlag: Springer International Publishing
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It may seem paradoxical at first, but a specific math problem can be harder to solve than some abstract generalization of it. For instance, if you want to know how many roots the equation
t
5
−
4
t
4
+
t
3
−
t
+
1
=
0
$$\displaystyle{t^{5} - 4t^{4} + t^{3} - t + 1 = 0}$$
can have then you could use calculus and figure it out. It would take a while. But thinking more abstractly, and with less work, you could show that every
n
th
-degree polynomial has at most
n
roots. In the same way many general results about functions of a real variable are more easily grasped at an abstract level—the level of metric spaces.