Uniformly Continuous Sets in Metric Spaces

1960 ◽  
Vol 67 (2) ◽  
pp. 153 ◽  
Author(s):  
Norman Levine ◽  
William G. Saunders
1960 ◽  
Vol 67 (2) ◽  
pp. 153-156 ◽  
Author(s):  
Norman Levine ◽  
William G. Saunders

1982 ◽  
Vol 47 (4) ◽  
pp. 772-792 ◽  
Author(s):  
Michael Katz

AbstractTwo real-valued deduction schemes are introduced, which agree on ⊢ ⊿ but not on Γ ⊢ ⊿, where Γ and ⊢ are finite sets of formulae. Using the first scheme we axiomatize real-valued equality so that it induces metrics on the domains of appropriate structures. We use the second scheme to reduce substitutivity of equals to uniform continuity, with respect to the metric equality, of interpretations of predicates in structures. This continuity extends from predicates to arbitrary formulae and the appropriate models have completions resembling analytic completions of metric spaces. We provide inference rules for the two deductions and discuss definability of each of them by means of the other.


2016 ◽  
Vol 17 (2) ◽  
pp. 123 ◽  
Author(s):  
Sejal Shah ◽  
Ruchi Das ◽  
Tarun Das

We prove that if a uniformly continuous self-map $f$ of a uniform space has topological specification property then the map $f$ has positive uniform entropy, which extends the similar known result for homeomorphisms on compact metric spaces having specification property. An example is also provided to justify that the converse is not true.<br /><br />


Filomat ◽  
2013 ◽  
Vol 27 (5) ◽  
pp. 925-930 ◽  
Author(s):  
Hüseyin Çakallı ◽  
Ayșe Sӧnmez

In this paper, we investigate slowly oscillating continuity in cone metric spaces. It turns out that the set of slowly oscillating continuous functions is equal to the set of uniformly continuous functions on a slowly oscillating compact subset of a topological vector space valued cone metric space.


1986 ◽  
Vol 29 (4) ◽  
pp. 463-468 ◽  
Author(s):  
Gerald Beer

AbstractLet 〈X, dx〉 and 〈Y, dY〉 be metric spaces and let hp denote Hausdorff distance in X x Y induced by the metric p on X x Y given by p[(x1, y1), (x2, y2)] = max ﹛dx(x1, x2),dY(y1, y2)﹜- Using the fact that hp when restricted to the uniformly continuous functions from X to Y induces the topology of uniform convergence, we exhibit a natural compactness criterion for C(X, Y) when X is compact and Y is complete.


1986 ◽  
Vol 104 ◽  
pp. 117-127 ◽  
Author(s):  
Takashi Shimomura

Let X be a compact space and f be a continuous map from X into itself. The topological entropy of f, h(f), was defined by Adler, Konheim and McAndrew [1]. After that Bowen [4] defined the topological entropy for uniformly continuous maps of metric spaces, and proved that the two entropies coincide when the spaces are compact. The definition of Bowen is useful in calculating entropy of continuous maps.


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