|
|
Lelkova hypotéza
Bartošová, Dana ; Simon, Petr (advisor) ; Balcar, Bohuslav (referee)
Lelek's conjecture which states that metric continua with span zero are chainable has been one of the most widely investigated problems in continuum theory over the past 40 years. We broaden our field of interest to non-metric continua and prove that if there is a non-metric counterexample to Lelek's conjecture we can convert it to a metric one. For a continuum X we take the lattice of all of its closed subsets 2X and consider a countable elementary sublattice L of 2X that we represent by a metric continuum wL via the Wallman representation for distributive lattices. By means of set theory, we obtain an L such that X is not chainable if and only if wL is not chainable and X has span zero if and only if wL has span zero. In the proof of the latter we use Shelah's theorem stating that every two elementarily equivalent models have isomorphic ultrapowers.
|
|
|
Vlastnosti metrických prostorů pomocí konvergence
Pokorný, Robin ; Hušek, Miroslav (advisor) ; Simon, Petr (referee)
In this thesis we deal with generalisation of the structure of convergence in metric spaces and characterisation of some properties using sequences. On basic of behaviour of convergent sequences in metric spaces which we alongside with selected properties in metric spaces remind we introduce the general structures. The first of them - sequential spaces - includes the information about the limit of a sequence which we consider to be unique. The second - uniformly sequential spaces - generalises the relation of adjacency of two sequences. We show that continuity of a mapping, topology, compactness, connectedness and separability can be induced from a sequential structure. In addition, we show total boundedness and completeness can be characterised using the term of a Cauchy sequence which we can define in a uniformly sequential structure. Boundedness is shown to be independent on both of those structures.
|
|
|
Products of Fréchet spaces
Olšák, Miroslav ; Simon, Petr (advisor) ; Chodounský, David (referee)
The article gives a constructions of k-tuples of topological spaces such that the product of the k-tuple is not Frchet-Urysohn but all smaller subproducts are. The construction uses almost disjoint systems. The article repeats the construction by Petr Simon of two such compact spaces. To achieve more dimensional example there are generalized terms of AD systems. The example is constructed under the assumption of existence of a strong completely separable MAD system. It is then constructed under the assumption s ≤ b where s is the splitting number and b is the bounding number.
|
|
| |
|
|
Topologies generated by adding single points
Bartoš, Adam ; Simon, Petr (advisor)
We define a general notion of closure scheme to systematically study the classes of Fréchet, sequetial, (pseudo)radial, (weakly) (discretely) Whyburn, and (weakly) discretely generated spaces. First, several general propositions on closure schemes and preservation of induced properties under topological constructions are proved and later applied when we systematically summarize the properties of the classes mentioned above. Next, we focus on a detailed overview of inclusions between the classes in the general case, in the case of Hausdorff spaces, and under additional conditions like compactness and countable compactness. Valid inclusions between the classes are summarized in well arranged diagrams, invalid inclusions are demonstrated by several counterexamples.
|
|
|
Products of Fréchet spaces
Olšák, Miroslav ; Simon, Petr (advisor) ; Chodounský, David (referee)
The article gives a constructions of k-tuples of topological spaces such that the product of the k-tuple is not Frchet-Urysohn but all smaller subproducts are. The construction uses almost disjoint systems. The article repeats the construction by Petr Simon of two such compact spaces. To achieve more dimensional example there are generalized terms of AD systems. The example is constructed under the assumption of existence of a strong completely separable MAD system. It is then constructed under the assumption s ≤ b where s is the splitting number and b is the bounding number.
|
|
|
Topologies generated by adding single points
Bartoš, Adam ; Simon, Petr (advisor)
We define a general notion of closure scheme to systematically study the classes of Fréchet, sequetial, (pseudo)radial, (weakly) (discretely) Whyburn, and (weakly) discretely generated spaces. First, several general propositions on closure schemes and preservation of induced properties under topological constructions are proved and later applied when we systematically summarize the properties of the classes mentioned above. Next, we focus on a detailed overview of inclusions between the classes in the general case, in the case of Hausdorff spaces, and under additional conditions like compactness and countable compactness. Valid inclusions between the classes are summarized in well arranged diagrams, invalid inclusions are demonstrated by several counterexamples.
|
|
|
Topologies generated by adding single points
Bartoš, Adam ; Simon, Petr (advisor) ; Hušek, Miroslav (referee)
We define a general notion of closure scheme to systematically study the classes of Fréchet, sequetial, (pseudo)radial, (weakly) (discretely) Whyburn, and (weakly) discretely generated spaces. First, several general propositions on closure schemes and preservation of induced properties under topological constructions are proved and later applied when we systematically summarize the properties of the classes mentioned above. Next, we focus on a detailed overview of inclusions between the classes in the general case, in the case of Hausdorff spaces, and under additional conditions like compactness and countable compactness. Valid inclusions between the classes are summarized in well arranged diagrams, invalid inclusions are demonstrated by several counterexamples.
|
|
|
Complete Boolean Algebras and Extremally Disconnected Compact Spaces
Starý, Jan ; Simon, Petr (advisor) ; Bukovský, Lev (referee) ; Thümmel, Egbert (referee)
We study the existence of special points in extremally disconnected compact topological spaces that witness their nonhomogeneity. Via Stone duality, we are looking for ultrafilters on complete Boolean algebras with special combinatorial properties. We introduce the notion of a coherent ultrafilter (coherent P-point, coherently selective). We show that generic existence of such ultrafilters on every complete ccc Boolean algebra of weight not exceeding the continuum is consistent with set theory, and that they witness the nonhomogeneity of the corresponding Stone spaces. We study the properties of the order-sequential property on σ-complete Boolean algebras and its relation to measure-theoretic properties. We ask whether the order-sequential topology can be compact in a nontrivial case, and partially answer the question in a special case of the Suslin algebra associated with a Suslin tree.
|
|
|
Connected compactifications
Vaváčková, Martina ; Simon, Petr (advisor) ; Vejnar, Benjamin (referee)
Title: Connected compactifications Author: Martina Vaváčková Department: Department of Theoretical Computer Science and Mathematical Logic Supervisor: prof. RNDr. Petr Simon, DrSc., Department of Theoretical Computer Science and Mathematical Logic Abstract: This thesis deals with connected compactifications of specific Tychonoff spaces. In particular, we are interested in the maximal elements with respect to the partial order over the set of all connected compactifications of a space. First we characterize maximal connected compactifications of spaces containing only finitely many components. We mention examples of spaces which have no connected compactification. Further we study connected compactifications of the rational numbers. We give a construction of a compactification analogical to the construction of the Čech-Stone compactification and we show a necessary and sufficient condition for its connectedness and maximality. Keywords: connected space, compact space, connectification, compactification
|
guest ::
RSS feed.