Weak Hausdorff space

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a weak Hausdorff space or weakly Hausdorff space is a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
where the image of every
continuous map In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
from a
compact Compact as used in politics may refer broadly to a pact A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ...

into the space is closed. 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. The notion was introduced by M. C. McCord. to remedy an inconvenience of working with the
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
of Hausdorff spaces. It is often used in tandem with
compactly generated space In topology, a compactly generated space (or k-space) is a topological space whose topology is coherent topology, coherent with the family of all compact space, compact subspaces. Specifically, a topological space ''X'' is compactly generated if it ...
s in
algebraic topology Algebraic topology is a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained ...
. A Δ-Hausdorff space is a topological space where the image of every
path A path is a route for physical travel - see Trail. Path or PATH may also refer to: Physical paths of different types * Bicycle path * Bridle path, used by people on horseback * Course (navigation), the intended path of a vehicle * Desire pat ...
is closed; i.e. if $f :$
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\to X is continuous, then $f\left($
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
is closed. Every weak Hausdorff space is Δ-Hausdorff, and every Δ-Hausdorff space is T1. A space is Δ-generated, if its topology is the finest such that each map $f : \Delta^n \to X$ from a topological n-simplex $\Delta^n$ to $X$ is continuous. Δ-Hausdorff spaces are to Δ-generated spaces as weak Hausdorff spaces are to compactly generated spaces.