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- Homework Statement
- show that T_{0} - space iff the derived set of every singleton is a union of closed sets.

- Relevant Equations
- T_{0} - space iff {x}^{'} is a union of closed sets.

Hello everyone,

Concerning the separation axioms in topology. Our topology professor introduced the equivalent definition for a topological space to be a ##T_{o}-space## as:

$$

(X,\tau)\ is\ a\ T_{o}-space\ iff\ \forall\ x\ \in X,\ \{x\}^{\prime}\ is\ a\ union\ of\ closed\ sets.

$$

The direction ##\implies## follows from the result ##\bar{\{x\}}\neq\bar{\{y\}}## for every distinct elements in ##T_{o}-space##: Let ##z\in \{x\}^{\prime}\implies z\neq x\implies x\notin\bar{\{z\}}##, since

$$

{z}\subseteq{x}^{\prime}\subseteq\bar{\{x\}}\implies \bar{\{z\}}\subseteq\bar{\{x\}}.

$$

Then

$$

\bar{\{z\}}=\bar{\{z\}}-\{x\}\subseteq\bar{\{x\}}-\{x\}=\{x\}^{\prime},

$$

thus ##z\in\bar{\{z\}}\subseteq\{x\}^{\prime}##, i.e., ##\{x\}^{\prime}## can be written as a union of closed sets. Unfortunately, the other direction is not that clear. Will appreciate any suggestions.

Concerning the separation axioms in topology. Our topology professor introduced the equivalent definition for a topological space to be a ##T_{o}-space## as:

$$

(X,\tau)\ is\ a\ T_{o}-space\ iff\ \forall\ x\ \in X,\ \{x\}^{\prime}\ is\ a\ union\ of\ closed\ sets.

$$

The direction ##\implies## follows from the result ##\bar{\{x\}}\neq\bar{\{y\}}## for every distinct elements in ##T_{o}-space##: Let ##z\in \{x\}^{\prime}\implies z\neq x\implies x\notin\bar{\{z\}}##, since

$$

{z}\subseteq{x}^{\prime}\subseteq\bar{\{x\}}\implies \bar{\{z\}}\subseteq\bar{\{x\}}.

$$

Then

$$

\bar{\{z\}}=\bar{\{z\}}-\{x\}\subseteq\bar{\{x\}}-\{x\}=\{x\}^{\prime},

$$

thus ##z\in\bar{\{z\}}\subseteq\{x\}^{\prime}##, i.e., ##\{x\}^{\prime}## can be written as a union of closed sets. Unfortunately, the other direction is not that clear. Will appreciate any suggestions.

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