Home - News ( United States | United Kingdom | Italy | Germany ) - Football scores

Showing content from https://en.wikipedia.org/wiki/Weak_Hausdorff_space below:

Weak Hausdorff space - Wikipedia

From Wikipedia, the free encyclopedia

In mathematics, a weak Hausdorff space or weakly Hausdorff space is a topological space where the image of every continuous map from a compact Hausdorff space into the space is closed.^[1] In particular, every Hausdorff space is weak Hausdorff. As a separation property, it is stronger than T₁, which is equivalent to the statement that points are closed. Specifically, every weak Hausdorff space is a T₁ space.^[2][3]

The notion was introduced by M. C. McCord^[4] to remedy an inconvenience of working with the category of Hausdorff spaces. It is often used in tandem with compactly generated spaces in algebraic topology. For that, see the category of compactly generated weak Hausdorff spaces.

A k-Hausdorff space^[5] is a topological space which satisfies any of the following equivalent conditions:

1. Each compact subspace is Hausdorff.
2. The diagonal { ( x , x ) : x ∈ X } {\displaystyle \{(x,x):x\in X\}} is k-closed in X × X . {\displaystyle X\times X.}
3. Each compact subspace is closed and strongly locally compact.

A Δ-Hausdorff space is a topological space where the image of every path is closed; that is, if whenever f : [ 0 , 1 ] → X {\displaystyle f:[0,1]\to X} is continuous then f ( [ 0 , 1 ] ) {\displaystyle f([0,1])} is closed in X . {\displaystyle X.} Every weak Hausdorff space is Δ {\displaystyle \Delta } -Hausdorff, and every Δ {\displaystyle \Delta } -Hausdorff space is a T₁ space. A space is Δ-generated if its topology is the finest topology such that each map f : Δ n → X {\displaystyle f:\Delta ^{n}\to X} from a topological n {\displaystyle n} -simplex Δ n {\displaystyle \Delta ^{n}} to X {\displaystyle X} is continuous. Δ {\displaystyle \Delta } -Hausdorff spaces are to Δ {\displaystyle \Delta } -generated spaces as weak Hausdorff spaces are to compactly generated spaces.

• Fixed-point space – Space where all functions have fixed points, a Hausdorff space where every continuous function from the space into itself has a fixed point.
• Hausdorff space – Type of topological space
• Locally Hausdorff space – Space such that every point has a Hausdorff neighborhood
• Particular point topology
• Quasitopological space – Function in topology
• Separation axiom – Axioms in topology defining notions of "separation"

1. ^ Hoffmann, Rudolf-E. (1979), "On weak Hausdorff spaces", Archiv der Mathematik, 32 (5): 487–504, doi:10.1007/BF01238530, MR 0547371.
2. ^ J.P. May, A Concise Course in Algebraic Topology. (1999) University of Chicago Press ISBN 0-226-51183-9 (See chapter 5)
3. ^ Strickland, Neil P. (2009). "The category of CGWH spaces" (PDF).
4. ^ McCord, M. C. (1969), "Classifying spaces and infinite symmetric products", Transactions of the American Mathematical Society, 146: 273–298, doi:10.2307/1995173, JSTOR 1995173, MR 0251719.
5. ^ Lawson, J; Madison, B (1974). "Quotients of k-semigroups". Semigroup Forum. 9: 1–18. doi:10.1007/BF02194829.

RetroSearch is an open source project built by @garambo | Open a GitHub Issue

Search and Browse the WWW like it's 1997 | Search results from DuckDuckGo

HTML: 3.2 | Encoding: UTF-8 | Version: 0.7.4