closed [2] if for every x 2 X and for every neighbourhood U of x there exists a neighbourhood V of x such that V ae U and f(V ) is closed. A space X is called hyperconnected [7] if every two nonempty open subsets of X have a nonempty intersection. Theorem. The following statements are equivalent for a T 1 space Y . (a) Y is compact. (b) Every function from a T 1 space into Y with all inverse images of compact sets closed is somewhat nearly continuous. (c) Every closed graph function from a T 1 space into Y is somewhat nearly continuous. (d)<F9

Abstract. We give a concrete example of a co-existential map between continua that is not confluent.

We propose a new method of combining ranking results which each rank the same set of items according to different criteria. It will choose a ranking that is closest as possible to each ranking result to be combined. In the context of the Internet, it can be used to rank web pages into an order that best reflects a balance between several criteria. In the context of sports, we propose to use the method to determine the winner of competitions, when the performance of an athlete is naturally judged according to different criteria, such as figure skating and show jumping. The rank aggregation method we propose is known to be NP-hard. This thesis aims to develop efficient algorithms to compute aggregated rankings for practically relevant instances. By employing parameterized complexity theory, we can identify the structural hardness of an instance and allow for choosing a high-performing algorithm accordingly. We present efficient and effective data reduction rules which will reduce parameters measuring the structural difficulty of an instance provably by simplifying the

This master’s thesis presents a number of important concepts in logic such as models, entailment, and proof calculi within the framework of category theory. By describing these concepts as categories, a tremendous amount of generality and power is gained. In particular, this approach makes it possible to reason about maps from one logic to another in a consistent and convenient manner. By a consistent map is meant that the truth stays invariant, that is, a statement true in the source logic is mapped to a similarly true statement in the target logic. Conversely, a statement false in the source logic is mapped to a statement false in the target logic. While the thesis focuses on the theoretical notions outlined above, a brief coverage of two practical applications is given as a means to illustrate the utility of these notions. Concluding the text is a chapter containing a discussion and a section wherein possible future work is presented. In an effort to make the text mostly self-contained, concepts beyond basic discrete mathematics are duly introduced with definitions and examples. These include, for

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