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I ntroduction

A mapping is said to be continuous, if two points close to each other remain to be close after the action of a mapping.

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xample 1. Consider the circle F and two points A, B at a short distance to each other. We chose a point C on the arc AB and tear the circle in the point C. We obtain an arc . New positions of the points A, B we denote A, B. Now A, B are far from each other. The mapping we constructed F is not continuous.

T he mapping is called homeomorphism or the topology mapping, if it is bijection and it is continuous in both sides. It means, that f and f^1are both continuous. The topology studies the properties of figures, which are preserved under the action of topology mappings. Two figures

F₁ and F₂ are said to be homeomorphic or topologically equivalent, if there exists a homeomorphism f:F₁F₂.

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xample 2. Denote F₁ a regular triangle, O  its center and F₂ its cir­cumscribed circle. Consider the central projection p:F₁F₂. This mapping is a homeomorphism (pic.2).

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xample 3. The surface of a cube and the surface of a cylinder are both topologically equivalent to a sphere. The topology mapping can be constructed as in example 2 (pic.3). All this surfaces are not homeomorphic to the surface of torus.

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pic.4

xample 3. The sphere is not homeomorphic to plain, but if we put out one point from the sphere, it becomes homeomorphic to the plain. The topology mapping is so called stereographic projection (pic.4).

One can imagine visually the topology mapping as follows: we can squeeze or stretch a set or crumple it, but you can’t tear or glue (paste) it.

The idea of continuity is the main idea of proves.

Example 4.

T heorem 1. A square can be circumscribed around any closed curve.

Proof. Consider a pair of parallel straight lines l and l, such that the curve  is located in the strip between them. Then we move this lines continuously until they become tangent to . The lines we get are called the lines of support for .

Lets draw one more pair lines of support m and m, that are perpendicular to l. We obtain the rectangle ABCD. Lets prove, that ABCD can be the square for some direction of the line l.

Let AD be the side, that are parallel to l and AB be the side, that are perpendicular to l. Denote the length of AD as h₁(l) and the length of AB as h₂(l). The circumscribed rectangle is a square, if h₁(l)h₂(l)=0.

Lets start to construct a rectangle with the pair of lines m and m. It coincides with ABCD. So

h₁(m)h₂(m)=(h₁(l)h₂(l)).

Lets turn the straight line l until it coincides with m. The circumscribed rectangle will be transformed continuously, so the value h₁(l)h₂(l) will vary continuously to the opposite value. Therefore there exists a position of the line l, when the value h₁(l)h₂(l) is equal to zero. It means, that there exists a position of the line l, when ABCD is a square.