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... 1. If X is a subspace of Y , then U = {V ∩ X : V ∈ V}. 2. X × Y is an Alexandroff space with minimal base U × V = {U × V : U ∈ U, V ∈ V}. These first two results are quoted without proof from [8]. Note that the property of Alexandroff is not countably productive, since discrete spaces are Alexandrof ...
... 1. If X is a subspace of Y , then U = {V ∩ X : V ∈ V}. 2. X × Y is an Alexandroff space with minimal base U × V = {U × V : U ∈ U, V ∈ V}. These first two results are quoted without proof from [8]. Note that the property of Alexandroff is not countably productive, since discrete spaces are Alexandrof ...
When does the Fell topology on a hyperspace of
... by Dolecki, Greco and Lechicki. These three authors showed that complete metric spaces are consonant. In our paper we give an example of a metric space with the Baire property which is not consonant. We also demonstrate that consonance is a delicate property by providing an example of two consonant ...
... by Dolecki, Greco and Lechicki. These three authors showed that complete metric spaces are consonant. In our paper we give an example of a metric space with the Baire property which is not consonant. We also demonstrate that consonance is a delicate property by providing an example of two consonant ...
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... semicontinuous. 2. Let f be the characteristic function of a set Ω ⊆ X. Then f is lower (upper) semicontinuous if and only if Ω is open (closed). This also holds for the function that equals ∞ in the set and 0 outside. It follows that the characteristic function of Q is not semicontinuous. 3. On R, ...
... semicontinuous. 2. Let f be the characteristic function of a set Ω ⊆ X. Then f is lower (upper) semicontinuous if and only if Ω is open (closed). This also holds for the function that equals ∞ in the set and 0 outside. It follows that the characteristic function of Q is not semicontinuous. 3. On R, ...
On sigma-Induced L-Fuzzy Topological Spaces
... Definition 1.1.[11]: A subset H of a topological space X is called a regular -subset if H is an intersection of a sequence of closed sets whose interiors contain H. Equivalently, if H = Gi = clXGi for i∈N ,where each Gi is open in X, then H is regular -subset of X . The complement of a regular -subs ...
... Definition 1.1.[11]: A subset H of a topological space X is called a regular -subset if H is an intersection of a sequence of closed sets whose interiors contain H. Equivalently, if H = Gi = clXGi for i∈N ,where each Gi is open in X, then H is regular -subset of X . The complement of a regular -subs ...
Topologies on Spaces of Subsets Ernest Michael Transactions of
... space X was taken by Hausdorff [lo, §28](4),who defined a metric on 2X in the case where X is a bounded metric space(5). If we extend the Hausdorff metric to d ( X ) , then sets with the same closure are a t zero distance from each other; hence the extended distance function is not a metric. When to ...
... space X was taken by Hausdorff [lo, §28](4),who defined a metric on 2X in the case where X is a bounded metric space(5). If we extend the Hausdorff metric to d ( X ) , then sets with the same closure are a t zero distance from each other; hence the extended distance function is not a metric. When to ...
Set-theoretic Problems Concerning Lindelöf Spaces
... We have already seen that the answer is positive under CH (Theorem 2.13), but if we add on definability conditions for the space, we can reduce that hypothesis to the existence of a Michael space: Theorem 3.5 ([Tala]). Every productively Lindelöf analytic metrizable space is σ-compact if and only ...
... We have already seen that the answer is positive under CH (Theorem 2.13), but if we add on definability conditions for the space, we can reduce that hypothesis to the existence of a Michael space: Theorem 3.5 ([Tala]). Every productively Lindelöf analytic metrizable space is σ-compact if and only ...
What to remember about metric spaces
... For a nonempty subset A of a metric space (X, d) its diameter is sup d(x, y) : x, y ∈ A . A nonempty set is bounded if its diameter is finite. A nonempty subset of a metric space is totally bounded if for every ε > 0, it can be covered by finitely many ε-balls. Boundedness and total boundedness are ...
... For a nonempty subset A of a metric space (X, d) its diameter is sup d(x, y) : x, y ∈ A . A nonempty set is bounded if its diameter is finite. A nonempty subset of a metric space is totally bounded if for every ε > 0, it can be covered by finitely many ε-balls. Boundedness and total boundedness are ...
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.