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# Lekcja CLIL Bartosz Kobyłecki  Proof, that if Klein Bottle is cut in half, two Mobius Strip are obtained.

 Question True False Reason Topology is any family of sets which is composed of defined on is a subset. x topology on a set X is a collection T of subsets of X, called open sets Each subset included in any any topology on a set is an open set. x A topology on a set X is a collection T of subsets of X, called open sets, satisfying: X and ∅ are open. The union of any family of open sets is open. Non-closed sets is called an open sets. x The members of τ are called open sets in X. A subset of X may be open, closed, both (a clopen set), or neither A circle is topologically equivalent to an ellipse. x A circle is topologically equivalent to an ellipse (into which it can be deformed by stretching) and a sphere is equivalent to an ellipsoid. A one to one, onto and continuous mapping between two topological spaces is called homeomorphism x A continuous deformation between a coffee mug and a donut (torus) illustrating that they are homeomorphic. But there need not be a continuous deformation for two spaces to be homeomorphic — only a continuous mapping with a continuous inverse function. Homeomorphism convert topological objects into another object in a continuous way withouth tearing and breaking, by bending. x The idea is that if one geometric object can be continuously transformed into another, then the two objects are to be viewed as being topologically the same. For example, a circle and a square are topologically equivalent. Real numbers, when considered toghether with the concept of distance on is an example of the  standard topological space. x The real numbers with the distance function given by the absolute difference, and, more generally, Euclidean n-space with the Euclideandistance, are complete metric spaces. The rational numbers with the samedistance function also form a metric space The set of rational number is countable x the set of rational numbersis countable Mobius Strip is surface that has two faces and one edge x An example of a Möbius strip can be created by taking a paper strip and giving it a half-twist, and then joining the ends of the strip to form a loop. Klein Bottle is bottle which has outside but not inside. x There is no inside or outside. If Klein Bottle is cut in half, two Mobius Strip are obtained. x

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