Locally Hausdorff space - AbsoluteAstronomy.com

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, in the field of topology

Topology

Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...

, a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

is said to be locally Hausdorff if every point has an open

Open set

The concept of an open set is fundamental to many areas of mathematics, especially point-set topology and metric topology. Intuitively speaking, a set U is open if any point x in U can be "moved" a small amount in any direction and still be in the set U...

neighbourhood that is Hausdorff

Hausdorff space

In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

under the subspace topology

Subspace topology

In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...

.

Here are some facts:

Every Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...

is locally Hausdorff.
Every locally Hausdorff space is T₁
T1 space
In topology and related branches of mathematics, a T1 space is a topological space in which, for every pair of distinct points, each has an open neighborhood not containing the other. An R0 space is one in which this holds for every pair of topologically distinguishable points...

.
There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space.
The bug-eyed line
Non-Hausdorff manifold
In mathematics, it is a usual axiom of a manifold to be a Hausdorff space, and this is assumed throughout geometry and topology: "manifold" means " Hausdorff manifold"....

is locally Hausdorff (it is in fact locally metrizable) but not Hausdorff.
The etale space for the sheaf
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...

of differentiable functions on a differential manifold is not Hausdorff, but it is locally Hausdorff.
A T₁ space need not be locally Hausdorff; an example of this is an infinite set given the cofinite topology.
Let X be a set given the particular point topology. Then X is locally Hausdorff at precisely one point. From the last example, it will follow that a set (with more than one point) given the particular point topology
Particular point topology
In mathematics, the particular point topology is a topology where sets are considered open if they are empty or contain a particular, arbitrarily chosen, point of the topological space. Formally, let X be any set and p ∈ X. The collectionof subsets of X is then the particular point topology...

is not a topological group
Topological group
In mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...

. Note that if x is the 'particular point' of X, and y is distinct from x, then any set containing y that doesn't also contain x inherits the discrete topology and is therefore Hausdorff. However, no neighbourhood of y is actually Hausdorff so that the space cannot be locally Hausdorff at y.
If G is a topological group that is locally Hausdorff at x for some point x of G, then G is Hausdorff. This follows from that fact that if y is a point of G, there exists a homeomorphism from G to itself carrying x to y, so G is locally Hausdorff at every point, and is therefore T₁ (and T₁ topological groups are Hausdorff).

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