Constructions and applications of rigid spaces, I

Kannan, V.; Rajagopalan, M.

doi:10.1016/0001-8708(78)90006-3

Cited by 24 publications

(13 citation statements)

References 17 publications

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“…This means that to obtain a counterexample, it suffices to have a strongly rigid space with a non-isolated point b such that at the same time {b} is an intersection of a family U i , i ∈ I, of open sets, with card I ≤ card X. Any of the examples of strongly rigid T 2 -spaces constructed in [5], [8] or [17] satisfies this condition. We next include an example of an AD-class which is not coreflective.…”

Section: A Subbase For This Topology Is Formed By the Setsmentioning

confidence: 99%

Hereditary, Additive and Divisible Classes in Epireflective Subcategories of Top

Sleziak¹

2006

Appl Categor Struct

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Abstract. Hereditary coreflective subcategories of an epireflective subcategory A of Top such that I 2 / ∈ A (here I 2 is the 2-point indiscrete space) were studied in [4]. It was shown that a coreflective subcategory B of A is hereditary (closed under the formation of subspaces) if and only if it is closed under the formation of prime factors. The main problem studied in this paper is the question whether this claim remains true if we study the (more general) subcategories of A which are closed under topological sums and quotients in A instead of the coreflective subcategories of A.We show that this is true if A ⊆ Haus or under some reasonable conditions on B. E.g., this holds if B contains either a prime space, or a space which is not locally connected, or a totally disconnected space or a non-discrete Hausdorff space.We touch also other questions related to such subclasses of A. We introduce a method extending the results from the case of non-bireflective subcategories (which was studied in [4]) to arbitrary epireflective subcategories of Top. We also prove some new facts about the lattice of coreflective subcategories of Top and ZD.

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Section: A Subbase For This Topology Is Formed By the Setsmentioning

confidence: 99%

Hereditary, Additive and Divisible Classes in Epireflective Subcategories of Top

Sleziak¹

2006

Appl Categor Struct

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“…Case 1: Let there exist a cardinal number n such that m = 2 n . Let X be the space constructed by c-process (see chapter 1 of [4] for the definition of this and related terms) from a c-system satisfying the following conditions:…”

mentioning

confidence: 99%

“…(This A* is the set of all points of X lying above some point of A. See § 1.1 in [4].) Then A* is a clopen subset of X.…”

mentioning

confidence: 99%

“…Then A* is a clopen subset of X. (See § 1.4. in [4].) Also, if A and B are distinct subsets of the first base space, then A* and B* are distinct.…”

mentioning

confidence: 99%

“…Since X satisfies some special conditions (see § 2.4. in [4]), 13X is zerodimensional. It is well-known that the map W -» W Pi X is a bijection between the family of all clopen subsets of @X and the family of all clopen subsets of X.…”

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confidence: 99%

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