229 NEARLY CONTINUOUS MULTIFUNCTIONS 1. Introduction
... Example 4. Define τF CT to be the collection consisting of ∅ together with all subsets of R whose complements in R are finite. Then, it is known that τF CT is a topology on R, called the finite complement topology. Take the discrete topology τD on R. We define the multifunction as follows; F : (R, τ ...
... Example 4. Define τF CT to be the collection consisting of ∅ together with all subsets of R whose complements in R are finite. Then, it is known that τF CT is a topology on R, called the finite complement topology. Take the discrete topology τD on R. We define the multifunction as follows; F : (R, τ ...
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... that the open intervals form a basis for the topology on R. Any open set in R can be expressed as a countable union of open intervals. Moreover, a set is closed if and only if it contains all its limit points. Definition 5.1. A subset A of R is called dense (in R) if any point x in R belongs to A or ...
... that the open intervals form a basis for the topology on R. Any open set in R can be expressed as a countable union of open intervals. Moreover, a set is closed if and only if it contains all its limit points. Definition 5.1. A subset A of R is called dense (in R) if any point x in R belongs to A or ...
f-9 Topological Characterizations of Separable Metrizable Zero
... A zero-dimensional space X is strongly homogeneous provided that all of its nonempty clopen sets are homeomorphic. It is easy to see that every strongly homogeneous space is homogeneous, [5, 1.9.3]. It is tempting to conjecture that all homogeneous zero-dimensional spaces are strongly homogeneous. T ...
... A zero-dimensional space X is strongly homogeneous provided that all of its nonempty clopen sets are homeomorphic. It is easy to see that every strongly homogeneous space is homogeneous, [5, 1.9.3]. It is tempting to conjecture that all homogeneous zero-dimensional spaces are strongly homogeneous. T ...
Consonance and Cantor set-selectors
... A metrizable space X is consonant if and only if its Vietoris hyperspace C(X ) is consonant. For a metrizable X , the Vietoris hyperspace C(X ) is also metrizable as mentioned above. If C(X ) is consonant, then each closed subset of it also is consonant [12, Proposition 4.2], see also Proposition 3. ...
... A metrizable space X is consonant if and only if its Vietoris hyperspace C(X ) is consonant. For a metrizable X , the Vietoris hyperspace C(X ) is also metrizable as mentioned above. If C(X ) is consonant, then each closed subset of it also is consonant [12, Proposition 4.2], see also Proposition 3. ...
A Topology Primer
... At first glance a notion as general as that of category seems of rather limited use. In fact, though, they turn out to be indispensable in more than one branch of mathematics as they allow to separate notions and results that are completely formal from those that are particular to a specific situati ...
... At first glance a notion as general as that of category seems of rather limited use. In fact, though, they turn out to be indispensable in more than one branch of mathematics as they allow to separate notions and results that are completely formal from those that are particular to a specific situati ...
Gruff ultrafilters - Centro de Ciencias Matemáticas UNAM
... In order for the previous theorem to be of any use, we need to exhibit models where ♦(rP ) holds. Recall that by [9, Thm. 6.6], in many of the models of Set Theory that are obtained via countable support iterations of proper forcing notions, we will have that ♦(rP ) holds if and only if rP = ω1 . Th ...
... In order for the previous theorem to be of any use, we need to exhibit models where ♦(rP ) holds. Recall that by [9, Thm. 6.6], in many of the models of Set Theory that are obtained via countable support iterations of proper forcing notions, we will have that ♦(rP ) holds if and only if rP = ω1 . Th ...