Topological Spaces with a Unique Compatible Quasi-Uniformity

Lindgren, William F.

doi:10.4153/cmb-1971-066-9

Cited by 16 publications

(14 citation statements)

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“…PROOF: It is well known that in a topological space that admits a unique quasiuniformity every interior-preserving open collection is finite ([5], Theorem 2.36) and as is observed in [5] p. 45, in order to establish the converse it suffices to show that a topological space in which every interior-preserving open collection is finite is a transitive space (see [5], Corollary 2.8 and [9], Corollary 3.5).…”

Section: Theorem: a Topological Space X Admits A Unique Quasi-uniformmentioning

confidence: 99%

“…The first part of Problem B is answered negatively in [7]; in this note we answer the second part positively. Topological spaces that admit a unique quasi-uniformity are considered in [1][2][3][4][5][8][9][10]. Related questions are dealt with in [6,7].…”

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confidence: 99%

“…It is known that a topological space that admits a unique quasi-uniformity is hereditarily compact ( [9]), [5] Theorem 2.36). The converse does not obtain ( [8], [5] Example 2.38).…”

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Topological Spaces with a Unique Compatible Quasi-Uniformity

Künzi

1986

Can. math. bull.

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Section: Theorem: a Topological Space X Admits A Unique Quasi-uniformmentioning

confidence: 99%

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Topological Spaces with a Unique Compatible Quasi-Uniformity

Künzi

1986

Can. math. bull.

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“…Proof. By Corollary 3.5 of [2] it is sufficient to show that <^Q is totally bounded. Hence the result will follow if we can show that for any g-cover # the number of distinct sets A^ (xeX) is finite.…”

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On topological spaces with a unique compatible quasi-uniformity

Brown

1977

Glasgow Math. J.

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It is shown in [2] that a uqu space satisfies the following conditions. {DC) There is no infinite, strictly decreasing sequence of open sets with open intersection. (IC) There is no infinite, strictly increasing sequence of open sets.In this note we show that for a transitive space these conditions are sufficient for the space to be uqu. This will follow as a consequence of the following result. THEOREM 1. Let y be a complete lattice of sets under the operations of intersection and union, in which all chains are finite. Then Sf is finite.Proof. This result is a consequence of standard theorems on lattice theory. See for example Corollary 2 of [1, p. 59] or Theorem 31 of [3, p. 116]. However, for the sake of completeness we give a direct proof. Since SP is a complete lattice we have for all ^e A^ u { S | S e y } . Also, for SeS?, 5 = v{A x \xeS} and so it will suffice that the number of distinct sets A x (xeX) is finite. We will call A a minimal\f xeA a implies A x = A a . Clearly distinct minimal sets are disjoint. Since all chains in £P are finite we see there are just finitely many distinct minimal sets, say A ai , ..., A Bn , that there exist a finite number of sets A bl , ..., A bm , which cover X, and that for any xeX there exist i,j with A ai £ A x £ A bj . Hence we may complete the proof by showing that if A a c A b then there are just finitely many distinct A x with A a £ A x £ A b . We will write A u < A v if A v is an immediate successor of A u for the ordering £ . Given A x c A y there exists zeX with A x < A Z Q A y , and hence there is a finite sequence A a < A., < ... < A Zt < A b , since £f contains no infinite chains. Moreover, for the same reason, the number of terms in such a sequence is bounded and each A x can only have a finite number of distinct immediate successors A z , so there are only finitely many distinct sequences A a < A 2l < ... < A, t < A b . Finally since each A x with A a c A X Q A b belongs to one of these sequences the proof is complete.An examination of the above proof shows that we have actually established the slightly more general result that if Sf is a set of sets, and if 9", the set of finite unions of the sets A x (xeX) is a subset of £f and contains no infinite chains, then y is finite.We return now to the question of uqu spaces. We recall that an open cover ( H> of the topological space (X, ST) is called a Q-cover if for all xeX. The quasi-uniformity <^Q with subbase is a Q-cover for (X,3T)} Glasgow Math. J. 18 (1977) 11-12.

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“…(Here we assume that D0 = X.) Such spaces are studied under the name upq spaces in [7,8] and in [3] it is observed that each uqp Hausdorff space is finite (compare also [1, Proposition 4]). Two simple countable examples of first-countable uqp 7 0 -spaces that are not hereditarily compact (see for example [9]) are provided in [4].…”

Section: Introductionmentioning

confidence: 99%

A nontrivial T₁-space admitting a unique quasi-proximity

Künzi¹,

Watson

1996

Glasgow Math. J.

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Abstract. We construct a r r space that is not hereditarily compact, although each of its open sets is the intersection of two compact open sets. The search for such a space was motivated by a problem in the theory of quasi-proximities. Introduction.Answering a question of [3] the first author showed in [4] that a topological space X admits a unique quasi-proximity if and only if its topology is the unique base that is closed under finite unions and finite intersections. (Here we assume that D0 = X.) Such spaces are studied under the name upq spaces in [7,8] and in [3] it is observed that each uqp Hausdorff space is finite (compare also [1, Proposition 4]). Two simple countable examples of first-countable uqp 7 0 -spaces that are not hereditarily compact (see for example [9]) are provided in [4]. In [5] it is observed that each uqp 7]-space in which each point is a G«-set is hereditarily compact. Similarly, locally hereditarily Lindelof uqp 7,-spaces are shown to be hereditarily compact in [2]. These results have led to the question whether each uqp 7,-space is hereditarily compact [2,6]. This simply stated problem has turned out to be very intractable. It is the aim of the present note to show that indeed a uqp 7i-space need not be hereditarily compact. Our example is scattered (i.e. each nonempty subspace contains an isolated point) and has the property that each of its open sets is the intersection of two compact open sets. The construction is carried out in ZFC, but the cardinality of the obtained space is very large. Curiously, it may be the largest space so far constructed in topology, without the use of additional set-theoretic axioms, to answer a specific question. The problem remains open whether each uqp 7J -space of small cardinality, say 2", may be hereditarily compact.

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Topological Spaces with a Unique Compatible Quasi-Uniformity

Cited by 16 publications

References 5 publications

Topological Spaces with a Unique Compatible Quasi-Uniformity

Topological Spaces with a Unique Compatible Quasi-Uniformity

On topological spaces with a unique compatible quasi-uniformity

A nontrivial T₁-space admitting a unique quasi-proximity

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Topological Spaces with a Unique Compatible Quasi-Uniformity

Cited by 16 publications

References 5 publications

Topological Spaces with a Unique Compatible Quasi-Uniformity

Topological Spaces with a Unique Compatible Quasi-Uniformity

On topological spaces with a unique compatible quasi-uniformity

A nontrivial T1-space admitting a unique quasi-proximity

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A nontrivial T₁-space admitting a unique quasi-proximity