Locally Hausdorff space

Locally Hausdorff space

In mathematics, in the field of topology, a topological space is said to be locally Hausdorff if every point has an open neighbourhood that is Hausdorff under the subspace topology.

Here are some facts:

  • Every Hausdorff space is locally Hausdorff.
  • Every locally Hausdorff space is T1.
  • There are locally Hausdorff spaces where a sequence has more than one limit. This can never happen for a Hausdorff space.
  • 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 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 is not a topological group. 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.
  • An infinite set in the finite complement topology is neither Hausdorff nor locally Hausdorff.
  • If G is a topological group that is locally Hausdorff at x for some point x of G, then G is locally Hausdorff at every point. 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

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