Skip to main content
MathOverflow

Return to Answer

added a Wikipedia link
Source Link
Martin Sleziak
Martin Sleziak
  • 4.7k
  • 4
  • 35
  • 40

Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space $X$ so that with one exception $x$, each point is open, but so that at the exceptional point, $X$ is not locally compact at $x$.

It is easy to find such a subspace of the real line. (Start with $0$ and$ (1/n)+(1/(m+n)$). Now delete each $1/n$).

Let $Y$ be the one point compactification, adding to $X$ a new point $y$, whose neighborhood complements are compact in $X$. In the new space $Y$, compact subsets are closed (and in particular $Y$ is WH), but $x$ and $y$ are inseparable.

See for example, Example 99 from Counterexamples in TopologyCounterexamples in Topology by Steen and Seebach.

Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space $X$ so that with one exception $x$, each point is open, but so that at the exceptional point, $X$ is not locally compact at $x$.

It is easy to find such a subspace of the real line. (Start with $0$ and$ (1/n)+(1/(m+n)$). Now delete each $1/n$).

Let $Y$ be the one point compactification, adding to $X$ a new point $y$, whose neighborhood complements are compact in $X$. In the new space $Y$, compact subsets are closed (and in particular $Y$ is WH), but $x$ and $y$ are inseparable.

See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.

Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space $X$ so that with one exception $x$, each point is open, but so that at the exceptional point, $X$ is not locally compact at $x$.

It is easy to find such a subspace of the real line. (Start with $0$ and$ (1/n)+(1/(m+n)$). Now delete each $1/n$).

Let $Y$ be the one point compactification, adding to $X$ a new point $y$, whose neighborhood complements are compact in $X$. In the new space $Y$, compact subsets are closed (and in particular $Y$ is WH), but $x$ and $y$ are inseparable.

See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.

Mathjax together with a ruthless distaste for unnecessary spacebar usage.
Source Link
Tyrone
Tyrone
  • 5.6k
  • 1
  • 29
  • 50

Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space X$X$ so that with one exception x$x$, each point is open, but so that at the exceptional point, X$X$ is not locally compact at x$x$.

It is easy to find such a subspace of the real line. ( startStart with 0 and (1/n)+(1/(m+n)$0$ and$ (1/n)+(1/(m+n)$) Now. Now delete each 1/n$1/n$).

Let Y$Y$ be the one point compactification, adding to X a$X$ a new point y$y$, whose neighborhood complements are compact in X$X$. In the new space Y$Y$, compact subsets are closed (and in particular Y$Y$ is WH), but x$x$ and y$y$ are inseparable.

See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.

Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space X so that with one exception x, each point is open, but so that at the exceptional point, X is not locally compact at x.

It is easy to find such a subspace of the real line. ( start with 0 and (1/n)+(1/(m+n)) Now delete each 1/n).

Let Y be the one point compactification, adding to X a new point y, whose neighborhood complements are compact in X. In the new space Y, compact subsets are closed (and in particular Y is WH), but x and y are inseparable.

See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.

Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space $X$ so that with one exception $x$, each point is open, but so that at the exceptional point, $X$ is not locally compact at $x$.

It is easy to find such a subspace of the real line. (Start with $0$ and$ (1/n)+(1/(m+n)$). Now delete each $1/n$).

Let $Y$ be the one point compactification, adding to $X$ a new point $y$, whose neighborhood complements are compact in $X$. In the new space $Y$, compact subsets are closed (and in particular $Y$ is WH), but $x$ and $y$ are inseparable.

See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.

Source Link
Paul Fabel
Paul Fabel
  • 2k
  • 15
  • 23

Here is perhaps the simplest example of a CGWH space which fails to be Hausdorff.

Start with a countable metric space X so that with one exception x, each point is open, but so that at the exceptional point, X is not locally compact at x.

It is easy to find such a subspace of the real line. ( start with 0 and (1/n)+(1/(m+n)) Now delete each 1/n).

Let Y be the one point compactification, adding to X a new point y, whose neighborhood complements are compact in X. In the new space Y, compact subsets are closed (and in particular Y is WH), but x and y are inseparable.

See for example, Example 99 from Counterexamples in Topology by Steen and Seebach.