Locally Hausdorff space

• Every Hausdorff space is locally Hausdorff.
• There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space.
• The line with two origins is locally Hausdorff (it is in fact locally metrizable) but not Hausdorff.
• The etale space for the sheaf 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 ${\displaystyle X}$ be a set given the particular point topology. Then ${\displaystyle X}$ 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 is not a topological group. Note that if ${\displaystyle x}$ is the 'particular point' of ${\displaystyle X,}$ and y is distinct from ${\displaystyle x,}$ then any set containing ${\displaystyle y}$ that doesn't also contain ${\displaystyle x}$ inherits the discrete topology and is therefore Hausdorff. However, no neighborhood of ${\displaystyle y}$ is actually Hausdorff so that the space cannot be locally Hausdorff at ${\displaystyle y.}$
• If ${\displaystyle G}$ is a topological group that is locally Hausdorff at some point ${\displaystyle x\in G,}$ then ${\displaystyle G}$ is Hausdorff. This follows from the fact that if ${\displaystyle y\in G,}$ there exists a homeomorphism from ${\displaystyle G}$ to itself carrying ${\displaystyle x}$ to ${\displaystyle y,}$ so ${\displaystyle G}$ is locally Hausdorff at every point, and is therefore T₁ (and T₁ topological groups are Hausdorff).

Every locally Hausdorff space is T₁.[2]

• Fixed-point space – Topological space such that every endomorphism has a fixed point, a Hausdorff space where every continuous function from the space into itself has a fixed point.
• Hausdorff space – Type of topological space
• Quasitopological space – a set X equipped with a function that associates to every compact Hausdorff space K a collection of maps K→C satisfying certain natural conditionsPages displaying wikidata descriptions as a fallback
• Separation axiom – Axioms in topology defining notions of "separation"
• Weak Hausdorff space – concept in algebraic topologyPages displaying wikidata descriptions as a fallback