Nonstratifiability of topological vector spaces

AbstractIt is shown that a compact K-metrizable space with a dense monotonically normal subspace is metrizable. It is deduced that if a Banach space, in its weak topology, is stratifiable, then it is metrizable. Also, it is shown that Cp(X) is stratifiable if and only if X is countable

Monotonicity in analytic topology

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Monotone normality in products

Monotone normality in finite and infinite topological products is investigated. As shown in (Heath et al., 1973), the countable (Tychonoff) power of a space is monotonically normal if and only if the space is stratifiable. It is shown that if the square of a space is monotonically normal, then all finite powers are monotonically normal and hereditarily paracompact. For certain special cases, it is observed that a space has all finite powers monotonically normal if and only if it linearly stratifiable. Nonetheless, a monotonically normal topological group is constructed, all of whose finite powers are monotonically normal, but which is not linearly stratifiable. The group is constructed using special filters and nonstandard topologies on infinite products

Monotone normality in products

Monotone normality in finite and infinite topological products is investigated. As shown in (Heath et al., 1973), the countable (Tychonoff) power of a space is monotonically normal if and only if the space is stratifiable. It is shown that if the square of a space is monotonically normal, then all finite powers are monotonically normal and hereditarily paracompact. For certain special cases, it is observed that a space has all finite powers monotonically normal if and only if it linearly stratifiable. Nonetheless, a monotonically normal topological group is constructed, all of whose finite powers are monotonically normal, but which is not linearly stratifiable. The group is constructed using special filters and nonstandard topologies on infinite products.Copyright 1999 Elsevier B.V. All rights reserved. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0

Elastic spaces may snap under perfect maps

The perfect image of an elastic space need not be elastic. Other relevant examples are presented

Diversity of p-adic analytic manifolds

AbstractThe problem of the cardinality of the set of non-homeomorphic p-adic manifolds is solved. It is proved that there exist 2ℵ1 pairwise non-homeomorphic non-metrizable one-dimensional p-adic analytic manifolds of weight ℵ1. This contrasts with the single isomorphism class of metrizable manifolds of the same weight. Further, we prove that for p>2, there are 2ℵ1 pairwise non-isomorphic non-metrizable manifolds of weight ℵ1, which are homeomorphic.To demonstrate the wide variety of non-metrizable p-adic manifolds, and contrast with the situation for real analytic manifolds, we construct a range of ‘pathological’ non-metrizable p-adic manifolds

Elastic spaces may snap under perfect maps

The perfect image of an elastic space need not be elastic. Other relevant examples are presented

Mal'tsev and retral spaces

A space X is Mal'tsev if there exists a continuous map M: X3 → X such that M(x, y, y) = x = M(y, y, x). A space X is retral if it is a retract of a topological group. Every retral space is Mal'tsev. General methods for constructing Mal'tsev and retral spaces are given. An example of a Mal'tsev space which is not retral is presented. An example of a Lindelöf topological group with cellularity the continuum is presented. Constraints on the examples are examined

Mal'tsev and retral spaces

A space X is Mal'tsev if there exists a continuous map M: X3 → X such that M(x, y, y) = x = M(y, y, x). A space X is retral if it is a retract of a topological group. Every retral space is Mal'tsev. General methods for constructing Mal'tsev and retral spaces are given. An example of a Mal'tsev space which is not retral is presented. An example of a Lindelöf topological group with cellularity the continuum is presented. Constraints on the examples are examined.Copyright 1997 Elsevier B.V. All rights reserved. Re-use of this article is permitted in accordance with the Terms and Conditions set out at http://www.elsevier.com/open-access/userlicense/1.0