slides
... Using only the operations in Σsw the subalgebra Σsw generated by the repeated application of operations in Σsw on B can approximate all spatial objects in C(X,R). Proof Idea: Localize the problem to a compact space K, find an approximate, and then show it works on C(X,R) •Pick any W(K,U) in C(X,R ...
... Using only the operations in Σsw the subalgebra Σsw generated by the repeated application of operations in Σsw on B can approximate all spatial objects in C(X,R). Proof Idea: Localize the problem to a compact space K, find an approximate, and then show it works on C(X,R) •Pick any W(K,U) in C(X,R ...
$\ alpha r $-spaces and some of their properties
... From definition of the sets A,, Bh i, j = 1, 2, 3, ... it follows that S'x and .T2 are complete r-systems in X and each r-system in X is either a part of &~\ or a part of ,T2. On the other hand it is clear that both pairs (X, S'I) and (X, ¥T2) are topological spaces, where .9~i and 5"2 are their cla ...
... From definition of the sets A,, Bh i, j = 1, 2, 3, ... it follows that S'x and .T2 are complete r-systems in X and each r-system in X is either a part of &~\ or a part of ,T2. On the other hand it is clear that both pairs (X, S'I) and (X, ¥T2) are topological spaces, where .9~i and 5"2 are their cla ...
To be published in Comment. Math. Univ. Carolinae CONTINUOUS
... and only if X is countable and discrete, when X is zero-dimensional and E is strongly zerodimensional and completely metrizable. In Section 4 we obtain our main results (Theorems 4.5 and 4.10): (1) For a zero-dimensional space X and a strongly zero-dimensional and metrizable space E, Cp(X, E) is wea ...
... and only if X is countable and discrete, when X is zero-dimensional and E is strongly zerodimensional and completely metrizable. In Section 4 we obtain our main results (Theorems 4.5 and 4.10): (1) For a zero-dimensional space X and a strongly zero-dimensional and metrizable space E, Cp(X, E) is wea ...
Instabilities of robot motion
... We will always assume that X is path connected, i.e., any pair of points A, B ∈ X may be joined by a continuous path γ in X. This means that it is possible to bring our system, by a continuous movement, from any given configuration A to any given configuration B. This assumption does not represent a ...
... We will always assume that X is path connected, i.e., any pair of points A, B ∈ X may be joined by a continuous path γ in X. This means that it is possible to bring our system, by a continuous movement, from any given configuration A to any given configuration B. This assumption does not represent a ...
General topology
In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set topology.The fundamental concepts in point-set topology are continuity, compactness, and connectedness: Continuous functions, intuitively, take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size. Connected sets are sets that cannot be divided into two pieces that are far apart. The words 'nearby', 'arbitrarily small', and 'far apart' can all be made precise by using open sets, as described below. If we change the definition of 'open set', we change what continuous functions, compact sets, and connected sets are. Each choice of definition for 'open set' is called a topology. A set with a topology is called a topological space.Metric spaces are an important class of topological spaces where distances can be assigned a number called a metric. Having a metric simplifies many proofs, and many of the most common topological spaces are metric spaces.