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My example is, $f : \mathbb{R}^+ \to \mathbb{R}$ defined by: $$f(x) = \begin{cases} x, &\text{if }0 \leq x < 1 \\ \tfrac{1}{x}, &\text{if }x \geq 1. \end{cases}$$ Even though $f(0)=0$ but $f([5,6]) \neq 1$. But $\mathbb{R}^+$ is a Hausdorff space but this continuous function fails to separate the point $0$ and $[5,6]$.

Is this correct ?

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  • $\begingroup$ Information about typesetting mathematics on this site can be found here. $\endgroup$ – user642796 Jul 30 '14 at 8:37
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    $\begingroup$ Just because one particular continuous function doesn't separate a point from a closed set doesn't mean that none do. For example the continuous function $f : \mathbb{R}^+ \to [0,1]$ defined by $f(x) = \min \{ x , 1 \}$ will separate $0$ from the closed interval $[5,6]$. (Note, too, that any subspace of a completely regular space is itself completely regular; since $\mathbb{R}$ is completely regular, so, too, is $\mathbb{R}^+$.) $\endgroup$ – user642796 Jul 30 '14 at 8:39
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A simple example of a Hausdorff space which is not (completely) regular is the K-topology on $\mathbb{R}$ using the set $K = \{ \frac 1n : n \geq 1 \}$. This is formed by taking as a base all open sets in the usual topology, as well as all sets of the form $(a,b) \setminus K$.

  • Since it is a finer topology, it is Hausdorff.

  • It is not too difficult to show that $K$ is a closed set in the new topology, and there are no disjoint open set $U,V$ with $0 \in U$ and $K \subseteq V$.

(Even finer grained examples of Hausdorff but not (completely) regular spaces may be found in the answers to my question here. It is easy to show that every regular (Hausdorff) space satisfies the criteria I was asking about.)

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$\pi$-Base is an online database of topological spaces inspired by Steen and Seebach's Counterexamples in Topology. It lists the following Hausdorff spaces that are not completely regular. You can learn more about the spaces by view the search result.

Alexandroff Plank

An Altered Long Line

Arens Square

Countable Complement Extension Topology

Deleted Diameter Topology

Deleted Radius Topology

Deleted Tychonoff Corkscrew

Double Origin Topology

Gustin's Sequence Space

Half-Disc Topology

Indiscrete Irrational Extension of $\mathbb{R}$

Indiscrete Rational Extension of $\mathbb{R}$

Irrational Slope Topology

Irregular Lattice Topology

Minimal Hausdorff Topology

Pointed Irrational Extension of $\mathbb{R}$

Pointed Rational Extension of $\mathbb{R}$

Prime Integer Topology

Rational Extension in the Plane

Relatively Prime Integer Topology

Roy's Lattice Space

Roy's Lattice Subspace

Simplified Arens Square

Smirnov's Deleted Sequence Topology

Strong Parallel Line Topology

Strong Ultrafilter Topology

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