T_{0} - Space Equivalent Definition

In summary: Suppose ##x \in \overline{\lbrace y \rbrace}##. Since ##\overline{\lbrace y \rbrace}## is closed, we have that ##x \in \overline{\lbrace y \rbrace} \cap \lbrace y \rbrace##, which contradicts the fact that ##\lbrace y \rbrace## is closed (because ##y## is not a limit point of ##\lbrace y \rbrace##). So, ##x \notin \overline{\lbrace y \rbrace}## and by the definition of ##T_0##, we can find an open set containing ##x## but not ##y##. This completes the proof.In summary, a ##T_0##-space is
  • #1
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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.
 
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  • #2
Sorry for the really dumb question, but is there an example where we can't replace the statement with: ##(X, \tau)## is ##T_0## if and only if for all ##x \in X##, ##\lbrace x \rbrace'## is empty? (I know ##\emptyset## is closed but...)

Definition: ##(X,\tau)## is ##T_0## if for any distinct points ##x,y \in X##, we can find an open set ##U## such that ##x \in U## and ##y \notin U##.

So, if ##y\neq x## was a limit point of ##\lbrace x \rbrace##, then by definition of ##T_0##, there is an open set ##U##such that ##y \in U## and ##(U - \lbrace y \rbrace) \cap \lbrace x \rbrace = \emptyset##. And this shows ##y## is not a limit point of ##\lbrace x \rbrace##.

And for any open set ##U## containing ##x##, we have ##(U - \lbrace x \rbrace) \cap \lbrace x \rbrace = \emptyset##, which shows ##x## is not a limit point of ##\lbrace x \rbrace##.

So ##\lbrace x \rbrace## has no limit points i.e., ##\lbrace x \rbrace'= \emptyset## ?
 
  • #3
fishturtle1 said:
Sorry for the really dumb question, but is there an example where we can't replace the statement with: ##(X, \tau)## is ##T_0## if and only if for all ##x \in X##, ##\lbrace x \rbrace'## is empty? (I know ##\emptyset## is closed but...)

Definition: ##(X,\tau)## is ##T_0## if for any distinct points ##x,y \in X##, we can find an open set ##U## such that ##x \in U## and ##y \notin U##.

So, if ##y\neq x## was a limit point of ##\lbrace x \rbrace##, then by definition of ##T_0##, there is an open set ##U##such that ##y \in U## and ##(U - \lbrace y \rbrace) \cap \lbrace x \rbrace = \emptyset##. And this shows ##y## is not a limit point of ##\lbrace x \rbrace##.

And for any open set ##U## containing ##x##, we have ##(U - \lbrace x \rbrace) \cap \lbrace x \rbrace = \emptyset##, which shows ##x## is not a limit point of ##\lbrace x \rbrace##.

So ##\lbrace x \rbrace## has no limit points i.e., ##\lbrace x \rbrace'= \emptyset## ?
I think I got the definition wrong. It should be (i think)

##(X,\tau)## is ##T_0## if for any distinct points ##x,y \in X##, we can find an open set ##U## such that ##x \in U## and ##y \notin U## or ##x \notin U## and ##y \in U##.

So in post #2, we can't guarantee there is an open set containing ##y## and not containing ##x##, which I assumed was possible... sorry. For the problem in OP, I think we can break it into cases:

Proof: ##(\Longleftarrow):## Let ##x, y## be distinct points in ##X## and consider two cases.

Case 1: ##x \in \lbrace y \rbrace'##. Then ##x \in C## for some closed set ##C \subset \lbrace y \rbrace'##. Since ##y## is not a limit point of ##\lbrace y \rbrace## (because ##(X - \lbrace y \rbrace) \cap \lbrace y \rbrace = \emptyset)##, we have that ##y \notin C##. So, ##y \in X - C## and ##x \notin X - C## where ##X - C## is open.

Case 2: ##x \notin \lbrace y \rbrace'##. Here we show ##x \notin \overline{\lbrace y \rbrace}## and then use a similar argument as in case 1.
 
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1. What is T0 - Space Equivalent Definition?

T0 - Space Equivalent Definition refers to the concept of a universal reference frame for measuring time and space. It is used in physics and astronomy to compare measurements from different locations and times.

2. How is T0 - Space Equivalent Definition calculated?

T0 - Space Equivalent Definition is calculated by taking into account the effects of time dilation and length contraction in special relativity. It is also affected by the curvature of space-time in general relativity.

3. Why is T0 - Space Equivalent Definition important in scientific research?

T0 - Space Equivalent Definition is important because it allows scientists to make accurate measurements and comparisons across different frames of reference. It also helps to reconcile the differences between classical and modern theories of physics.

4. How does T0 - Space Equivalent Definition impact our understanding of the universe?

T0 - Space Equivalent Definition plays a crucial role in our understanding of the universe by providing a consistent framework for measuring and describing the behavior of matter and energy. It also helps to explain the observed phenomena such as time dilation and the expansion of the universe.

5. Are there any limitations to T0 - Space Equivalent Definition?

While T0 - Space Equivalent Definition is a useful concept, it is not without its limitations. It does not account for the effects of gravity on time and space, and it is only valid in the absence of significant gravitational fields. Additionally, it does not provide a complete understanding of the nature of space and time, and further research is needed to fully comprehend these concepts.

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