On the other hand, in a non-Hausdorff space, a constant sequence (which obviously converges) can have infinitely many limits! Typical examples. Certainly homotopy theory (up to weak homotopy equivalence) needs …

Any discrete space (i.e., a topological space with the discrete topology) is a Hausdorff space. Formal definition. Robert Goldblatt realized in a 2012 paper that the Rasiowa-Sikorski lemma is a consequence of the fact that all compact Hausdorff spaces are Baire (plus some Stone duality, which is the main bridge between logic and topology). : Univ., Bern Sponsoring Org.

The property of topological spaces of being a Hausdorff space does not imply the property of being a regular space.. Verbal statement. • Every compact subspace of a Hausdorff space is closed. Let X be a topological space.Most commonly X is called locally compact, if every point of X has a compact neighbourhood.. This is a beautiful argument… read the full post. On non-Hausdorff spaces Ivan L. Reilly Department qf Mathematics and Statistics, University of Auckland, Auckland, New Zealand Received 30 October 1989 Revised 3 April 1990 and 11 June 1990 Abstract Reilly, I.L., On non-Hausdorff spaces, Topology and its Applications 44 (1992) 331-340. A non-Hausdorff topological space is a topological space which is not a Hausdorff topological space. Examples. Non-Hausdorff manifold Last updated June 28, 2019.

In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to Euclidean space, but not necessarily Hausdorff.

Authors: Hajicek, P Publication Date: Fri Jan 01 00:00:00 EST 1971 Research Org. Hausdorff spaces are a kind of nice topological space; they do not form a particularly nice category of spaces themselves, but many such nice categories consist of only Hausdorff spaces. • In a Hausdorff space, any point and a disjoint compact subspace can be separated by open sets, in the sense that there exist disjoint open sets, and one contains the point and the other contains the compact subspace.

Suppose that X is a locally compact Hausdorff space. (The etale space is Hausdorff if it is a sheaf of functions with some sort of analytic continuation property.) One can say more. One example is the space constructed by Munkres. for every non-negative number: it has a "fork" at zero.

A topological space (X, 3) is called R0 if x E U E 9 implies cl(x) c CJ, where cl A denotes the F-closure of A. • Every subspace of a \$\${T_2}\$\$ space is a \$\${T_2}\$\$ space.

1. every point of X has a compact neighbourhood.

If a subset of a metric space (X, d) is compact then it is d-bounded. There are several other interesting classes of non-Hausdorff spaces which should capture our attention.

The etale space of a sheaf, such as the sheaf of continuous real functions over a manifold, is a manifold that is often non-Hausdorff. That is Proposition 8.3.24 in the book. Regularity is not refining-preserved. Details are provided in .

Etale space.

The étale space construction produces non-Hausdorff and nonparacompact spaces (e.g., smooth manifolds) in many practical examples that have nothing to do with algebraic geometry. This paper surveys some recent work on topological spaces which need not satisfy the Hausdorff … : USDOE OSTI Identifier: 4720453 NSA Number: In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space.