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# Lekcja CLIL (Szymon Kwaśniewski)

 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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