Volume 34, Issue 1, January 2022, 101713
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T3 Regular Hausdorff Topological Spaces
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Table of Contents
T3 Regular Hausdorff Topological Spaces |
Recall from the T0 Kolmogorov Topological Spaces page that a topological space $X$ is said to be a T0 space or a Kolmogorov space if for every pair of distinct points $x, y \in X$, $x \neq y$ there exists open neighbourhoods $U$ of $x$ and $V$ of $y$ such that either $x \not \in V$ or $y \not \in U$.
On the T1 Fréchet Topological Spaces page we said that a topological space $X$ is said to be a T1 space or a Fréchet space if for every pair of distinct points $x, y \in X$, $x \neq y$ there exists open neighbourhoods $U$ of $x$ and $V$ of $y$ such that $x \not \in V$ and $y \not \in U$.
On the T2 Hausdorff Topological Spaces page we said that a topological space $X$ is said to be a T2 space or a Hausdorff space if for every pair of distinct points $x, y \in X$, $x \neq y$ there exists open neighbourhoods $U$ of $x$ and $V$ of $y$ such that $U \cap V = \emptyset$.
Before we look at the next type of topological space, we will first need to give some definitions.
| Definition: Let $X$ be a topological space and let $A$ and $B$ be disjoint subset of $X$, i.e., $A \cap B = \emptyset$. If $U$ and $V$ are open sets such that $A \subseteq U$, $B \subseteq V$, and $U \cap V = \emptyset$ then $U$ and $V$ are said to Separate $A$ and $B$. A topological space $X$ is said to be Regular if for every $x \in X$ and for every closed set $F \subseteq X$ not containing $x$ there exists open sets $U$ and $V$ which separate $\{ x \}$ and $F$. |
We will now define the next type of topological space called a T3 or Regular Hausdorff topological space.
| Definition: Let $X$ be a topological space. Then $X$ is said to be a T3 Space or a Regular Hausdorff Space if $X$ is both a regular space and a T1 space.
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| Theorem 1: Let $X$ be a topological space. If $X$ is a T3 space then $X$ is a T2 space. |
- Proof: Let $X$ be a T3 space. Then $X$ is both a regular space and a T1 space. Let $x, y \in X$, $x \neq y$. Since $X$ is a T1 space, every singleton set is closed. In other words, $\{ x \}$ and $\{ y \}$ are both closed. Since $X$ is a regular space, if we let $F = \{ y \}$ then there exists open sets $U$ of $\{ x \}$ and $V$ of $F = \{ y \}$ such that $U \cap V = \emptyset$. So $X$ is a T2 space. $\blacksquare$
Let $$\left[ {X,\tau } \right]$$ be a topological space, then for every non-empty closed set $$F$$ and a point $$x$$ which does not belong to $$F$$, there exist open sets $$U$$ and $$V$$, such that $$x \in U,{\text{ }}F \subseteq U$$ and $$U \cap V = \phi $$.
In other words, a topological space $$X$$ is said to be a regular space if for any $$x \in X$$ and any closed set $$A$$ of $$X$$, there exist open sets $$U$$ and $$V$$ such that $$x \in U,{\text{ A}} \subseteq U$$ and $$U \cap V = \phi $$.
Example:
Show that a regular space need not be a Hausdorff space.
For this, let $$X$$ be an indiscrete topological space, then the only non-empty closed set is $$X$$, so for any $$x \in X$$, there does not exist a closed set $$A$$ which does not contain $$x$$. so $$X$$ is trivially a regular space. Since for any $$x,y \in X,\;x \ne y$$, there is only one open set $$X$$ itself containing these points, so $$X$$ is not a Hausdorff space.
T3-Space
A regular $${T_1}$$ space is called a $${T_3}$$ space.
Theorems • Every subspace of a regular space is a regular space. • Every $${T_3}$$ space is a Hausdorff space.
• Let $$X$$ be a topological space, then the following statements are equivalent: [1] $$X$$ is a regular space; [2] for every open set $$U$$ in $$X$$ and a point $$a \in U$$ there exists an open set $$V$$ such that $$a \in V \subseteq \overline V \subseteq U$$; [3] every point of $$X$$ has a local neighborhood basis consisting of closed sets.
From Topospaces
A topological space is termed a regular Hausdorff space or a
Note that outside of point-set topology, and in many elementary treatments, the term regular space is used to stand for regular Hausdorff space.
This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces
In the T family [properties of topological spaces related to separation axioms], this is called: T3
Relation with other properties
Stronger properties
| metrizable space | underlying topological space of a metric space | Tychonoff space|FULL LIST, MORE INFO | ||
| CW-space | topological space arising as the underlying space of a CW-complex | |FULL LIST, MORE INFO | ||
| perfectly normal Hausdorff space | it is normal and every closed subset is a G-delta subset | |FULL LIST, MORE INFO | ||
| hereditarily normal Hausdorff space | every subset is normal in the subspace topology | |FULL LIST, MORE INFO | ||
| monotonically normal Hausdorff space | [follow link for definition] | |FULL LIST, MORE INFO | ||
| normal Hausdorff space | T1 and disjoint closed subsets can be separated by disjoint open subsets | Tychonoff space|FULL LIST, MORE INFO | ||
| Tychonoff space | T1 and point and closed subset not containing it can be separated by continuous function | |FULL LIST, MORE INFO | ||
| compact Hausdorff space | compact and Hausdorff | Tychonoff space|FULL LIST, MORE INFO | ||
| locally compact Hausdorff space | locally compact and Hausdorff | |FULL LIST, MORE INFO | ||
| paracompact Hausdorff space | |FULL LIST, MORE INFO |