TIETZE Extension Theorem Wiki

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1 Motivation
Mathematics could be described as an ultimate quest for generalization. Everyone knows that in a (3-4-5) right triangle,
32 + 42 = 52:
As Pythagoras knew, however, this is the case for any right triangle. That is, the sum of the squares of the legs equals the square on the hypotenuse. Or as an equation,
a2 + b2 = c2:
This of course can handle a much broader range of triangles, but it is still not satisfactory enough.
This formula gives no information about triangles that are not right, so this formula was later
generalized to the so-called \Law of Cosines"
a2 = b2 + c2 ¡ 2bc cosA

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TIETZE Extension Theorem.pdf (110.7 KB, 26 views)

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As you need TIETZE Extension Theorem, so I am giving complete details about TIETZE Extension Theorem:

TIETZE Extension Theorem
Motivation
Mathematics could be described as an ultimate quest for generalization. Everyone knows that in a
(3-4-5) right triangle,

¬¬ 32+42=52

This of course can handle a much broader range of triangles, but it is still not satisfactory enough.
This formula gives no information about triangles that are not right, so this formula was later
generalized to the so-called \Law of Cosines"

This formula can be applied to any triangle that you can draw. This quest towards generalization
is a good way to think of mathematics, and a good way to lead into this short topological paper.
Although the above example is quite elementary, in a way, so is the question behind the Tietze
Extension Theorem. Take for instance the curve de¯ned by f(x) = x on the closed interval [0; 1].
The graph of this curve is a line segment. Indeed, what we really have is a continuous function
(hereafter called a map):

Basic Preliminaries
A background in topology will undoubtedly be needed to get the most out of this paper, but in an
attempt to make this paper accessible to all readers I will brie°y de¯ne all pertinent terms.
A topology on a set X is a family of subsets T such that the following properties hold:
1 Both the empty set and the whole set X belong to T 2 The union of any collection of sets in T belongs to T .
3 The intersection of a ¯nite collection of sets in T belongs to T

TIETZE Extension Theorem

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