I need help in order to fully understand the order topology ... and specifically Example 1.4.4 ... ...

Example 1.4.4 reads as follows:

Singh - Example 1.4.4 ... Ordered Space .png

In order to fully understand Example 1.4.4 I decided to take $ \displaystyle X = \{ a, b, c \}$ where $ \displaystyle a \leq b, a \leq c$ and $ \displaystyle b \leq c$ ... ...

Now in the above text, Singh writes the following:

" ... ... The basis generated by the subbasis of $ \displaystyle X$ consists of all open rays, all open intervals $ \displaystyle (a, b)$, the emptyset $ \displaystyle \emptyset$, and the full space $ \displaystyle X$. ... ... "

Now as I understand it the open rays in $ \displaystyle X$ are as follows:

$ \displaystyle ( - \infty, a) = \emptyset$

$ \displaystyle ( - \infty, b) = \{ a \}$

$ \displaystyle ( - \infty, c) = \{ a, b \}$

$ \displaystyle ( a, \infty) = \{ b, c \}$

$ \displaystyle ( b, \infty) = \{ c \}$

$ \displaystyle ( c, \infty) = \emptyset$

... and (see definition of order topology below) the open rays constitute the subbasis of the order topology ...

To generate the basis, according to the text of Example 1.4.4, we have to add in all open intervals $ \displaystyle (a, b)$, the emptyset $ \displaystyle \emptyset$, and the full space $ \displaystyle X$. ... ...

The open intervals in $ \displaystyle X$ are as follows:

$ \displaystyle (a, b) = \emptyset$

$ \displaystyle (b, c) = \emptyset$

$ \displaystyle (a, c) = \{ b \}$

The above open rays, open intervals together with $ \displaystyle \emptyset$ (already in the basis) and $ \displaystyle X$ constitute a basis for the order topology of the ordered set $ \displaystyle X$ ... ...

Can someone please confirm that the above analysis is correct and/or point out errors or shortcomings ... ..

Help will be much appreciated ... ...

Peter

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It may help MHB readers of the above post to have access to Singh's definitions of the order topology, together with the definitions of subbasis and basis ... so I am providing the same ... as follows:

Singh - Defn 1.4.2 ... Order Topology ... .png

Singh - Start of Sectio 1.4 ... .png

Singh - Defn 1.4.3 ... ... Basis .png

Hope that helps ... ...

Peter

I need help in order to fully understand some remarks by Singh just before he defines a sub-basis ... ..

The relevant text reads as follows:

Singh - Start of Sectio 1.4 ... .png

I am unsure of Singh's arguments concerning the nature of $ \displaystyle \mathcal{ T } ( \mathcal{S} )$ ... ...

Singh writes the following:

" ... ... Clearly $ \displaystyle \mathcal{ T } ( \mathcal{S} )$ is the coarsest topology on X containing $ \displaystyle \mathcal{S} $. It consists of $ \displaystyle \emptyset, X$, all finite intersections of members of $ \displaystyle \mathcal{S}$ and all unions of these finite intersections. This can be easily be ascertained by verifying that the collection of these sets is a topology for X, which contains $ \displaystyle \mathcal{S} $ and is coarser than $ \displaystyle \mathcal{ T } ( \mathcal{S} )$. ... ... "

My questions are as follows:

Why exactly is the collection of sets specified necessarily coarser than $ \displaystyle \mathcal{S}$ ... ?

Indeed ... the comment is confusing since Singh appears to say that the collection of sets mentioned

Can someone please clarify Singh's argument ...

Help will be much appreciated ...

Peter

I need help in order to fully understand some remarks by Singh just before he defines a sub-basis ... ..

The relevant text reads as follows:

Singh - Start of Sectio 1.4 ... .png

To try to fully understand the above text by Singh I tried to work the following example:

$ \displaystyle X = \{ a, b, c \}$ and $ \displaystyle \mathcal{S} = \{ \{ a \}, \{ b \} \}$

Topologies containing $ \displaystyle \mathcal{S}$ are as follows:

$ \displaystyle \mathcal{ T_1 } = \{ X, \emptyset, \{ a, b \} , \{ a, c \}, \{ b, c \}, \{ a \}, \{ b \}, \{ c \} \}$

$ \displaystyle \mathcal{ T_2 } = \{ X, \emptyset, \{ a, b \} , \{ a, c \}, \{ a \}, \{ b \} \}$

$ \displaystyle \mathcal{ T_3 } = \{ X, \emptyset, \{ a, b \} , \{ b, c \}, \{ a \}, \{ b \} \}$

$ \displaystyle \mathcal{ T_4 } = \{ X, \emptyset, \{ a, b \} , \{ a \}, \{ b \} \}$

Therefore $ \displaystyle \mathcal{ T } ( \mathcal{S} ) = \mathcal{ T_1 } \cap \mathcal{ T_2 } \cap \mathcal{ T_3 } \cap \mathcal{ T_4 }$

$ \displaystyle = \{ X, \emptyset, \{ a, b \} , \{ a \}, \{ b \} \}$

But ... now Singh writes the following ...

" ... Clearly $ \displaystyle \mathcal{ T } ( \mathcal{S} )$ is the coarsest topology. It consists of $ \displaystyle \emptyset, X$, all finite intersections of members of $ \displaystyle \mathcal{S}$ and all unions of these finite intersections. ... ..."

However ... all finite intersections of members of $ \displaystyle \mathcal{S}$ comprises $ \displaystyle \{ a \} \cap \{ b \} = \emptyset$ ... and so, b this reckoning ... $ \displaystyle \mathcal{ T } ( \mathcal{S} )$ consists of $ \displaystyle X$ and $ \displaystyle \emptyset$ ...

Can someone clarify the above ...

Peter

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There is a small fragment of relevant text in Singh Section 1.2 ... it reads as follows:

Singh - Propn 1.2.2 ... .png

Hope that helps ... ...

Peter

I need some fuuther help in order to fully understand Example 4.1.1 ...

Example 4.1.1 reads as follows:

Singh - Example 4.1.1 ... .png

In the above example from Singh we read the following:

" ... ...Then the complement of $ \displaystyle \{ x_n \ | \ x_n \neq x \text{ and } n = 1,2, ... \}$ is a nbd of $ \displaystyle x$. Accordingly, there exists an integer $ \displaystyle n_0$ such that $ \displaystyle x_n = x$ for all $ \displaystyle n \geq n_0$. ... ... "

My question is as follows:

Why, if the complement of $ \displaystyle \{ x_n \ | \ x_n \neq x$ and $ \displaystyle n = 1,2, ... \}$ is a nbd of $ \displaystyle x$ does there exist an integer $ \displaystyle n_0$ such that $ \displaystyle x_n = x$ for all $ \displaystyle n \geq n_0$. ... ... ?

Help will be much appreciated ... ...

Peter

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It may help readers of the above post to have access to Singh's definition of a neighborhood and to the start of Chapter 4 (which gives the relevant definitions) ... so I am providing the text as follows:

Singh - Defn 1.2.5 ... ... NBD ... .png

Singh - 1 - Start of Chapter 4 ... PART 1 .png

Singh - 2 - Start of Chapter 4 ... PART 2 .png

Hope that helps ...

Peter

I need help in order to fully understand Example 4.1.1 ...

Example 4.1.1 reads as follows:

Singh - Example 4.1.1 ... .png

In the above example from Singh we read the following:

" ... ...no rational number is a limit of a sequence in $ \displaystyle \mathbb{R} - \mathbb{Q}$ ... ... "

My question is as follows:

Why exactly is it the case that no rational number a limit of a sequence in $ \displaystyle \mathbb{R} - \mathbb{Q}$ ... ... "

Help will be appreciated ...

Peter

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It may help readers of the above post to have access to Singh's definition of a neighborhood and to the start of Chapter 4 (which gives the relevant definitions) ... so I am providing the text as follows:

Singh - Defn 1.2.5 ... ... NBD ... .png

Singh - 1 - Start of Chapter 4 ... PART 1 .png

Singh - 2 - Start of Chapter 4 ... PART 2 .png

Hope that helps ...

Peter

I need help in order to fully understand Singh's proof of Theorem 1.3.10 ...

Theorem 1.3.10 (plus the definition of boundary) reads as follows:

Singh - Defn 1.3.9 and Theorem 1.3.10 ... .png

In the above proof by Singh we read the following:

" ... ... Conversely, if $ \displaystyle x \in \overline{A} - A$, then $ \displaystyle x \in \overline{A} \cap (X - A) \subseteq \partial A$ and the reverse inclusion follows. ... ... "

My questions are as follows:

Why is it true that $ \displaystyle \overline{A} \cap (X - A) \subseteq \partial A$ ... ?

I suspect that this is because $ \displaystyle (X - A) \subseteq \overline{ (X - A) }$ ... ... is that correct ... ?

How does $ \displaystyle x \in \overline{A} \cap (X - A) \subseteq \partial A$ lead to the reverse inclusion being true ... ... ?

(I am assuming that the reverse inclusion is $ \displaystyle \overline{A} \subseteq A \cup \partial A$ ... )

Help will be much appreciated ... ...

Peter

I need help in order to formulate a rigorous proof of

Proposition 1.3.2 reads as follows:

Singh - Propn 1.3.2 ... .png

Can some please help me to formulate a formal and rigorous proof of Proposition 1.3.2 (a) ... that is to formally and rigorously demonstrate that $ \displaystyle (A^{ \circ })^{ \circ } = A^{ \circ }$ ... using only Definition 1.3.1 ...

Help will be much appreciated ...

Peter

I need help in order to fully understand Singh's proof of Theorem 1.3.7 ... (using only the definitions and results Singh has established to date ... see below ... )

Theorem 1.3.7 reads as follows:

Singh - 1 - Theorem 1.3.7 ... PART 1 ... .png

Singh - 2 - Theorem 1.3.7 ... PART 2 ... .png

In the above proof by Singh we read the following:

" ... ... So $ \displaystyle U$ is contained in the complement of $ \displaystyle A \cup A'$, and hence $ \displaystyle A \cup A'$ is closed. It follows that $ \displaystyle \overline{A} \subseteq A \cup A'$ ... ... "

My question is as follows:

Why does $ \displaystyle U$ being contained in the complement of $ \displaystyle A \cup A'$ imply that $ \displaystyle A \cup A'$ is closed ... and further why then does it follow that $ \displaystyle \overline{A} \subseteq A \cup A'$ ... ...

Help will be appreciated ...

Peter

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It is important that any proof of the above remarks only rely on the definitions and results Singh has established to date ... namely Definition 1.3.3, Proposition 1.3.4, Theorem 1.3.5 and Definition 1.3.6 ... which read as follows ... :

Singh - 1 - Defn 1.3.3, Propn 1.3.4, Theorem 1.3.5, Defn 1.3.6 ... PART 1 ... .png

Singh - 2 - Defn 1.3.3, Propn 1.3.4, Theorem 1.3.5, Defn 1.3.6 ... PART 2 ... .png

Hope that helps ...

Peter

I need some help in order to fully understand Kalajdzievski's definition of a closed set in a topological space ...

The relevant text reads as follows:

K - Defn of a Closed Subset of a Toplogical Space ... .png

As I understand it many closed subsets of the underlying set $ \displaystyle X$ of a topological space $ \displaystyle (X, \tau)$ do not belong to the topological space because they are not open ... i.e. they are not clopen sets ...

Is my interpretation of the above situation correct ... ... ?

Help will be appreciated ...

Peter

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It may help readers of the above post to have available Kalajdzievski's definition of a topological space ... so I am providing the same ... as follows:

K - Defn of a Topological Space ... .png

I am currently focused on Chapter 8: Continuity in Topological Spaces; bases ...

I need some help in order to prove Definition 8.1 is essentially equivalent to Definition 8.2 ... ...

Definitions 8.1 and 8.2 read as follows: ... ...

Sutherland - Defn 8.1 and Defn 8.2 ... .png

In the above text we read the following:

" ... ... Then one can prove that $ \displaystyle f$ is continuous iff it is continuous at every point of $ \displaystyle X$. ... ... "

I sketched out a proof of the above statement ... but am unsure of the correctness/validity of my proof ...

My sketch of the proof is as follows ...

We are given (Definition 8.2) that $ \displaystyle U' \in T_Y$ where $ \displaystyle f(x) \in U'$ ... ...

Take $ \displaystyle U = f^{-1} (U')$

Then $ \displaystyle x \in U$ ...

Also from Definition 8.1 we have $ \displaystyle U \in T_X$ ...

and further $ \displaystyle f(U) = f(f^{-1}(U')) \subseteq U'$ ...

... that is Definition 8.2 holds at any $ \displaystyle x \in X$ ...

Let $ \displaystyle V \in T_Y$ ... need to show $ \displaystyle f^{-1} (V) \in T_X$ ... ...

Now $ \displaystyle x \in f^{-1} (V) \Longrightarrow f(x) \in V$

$ \displaystyle \Longrightarrow$ there exists a set $ \displaystyle U_x \in T_X$ such that $ \displaystyle f(U_x) \subseteq V$ by Definition 8.2 ...

But $ \displaystyle f(U_x) \subseteq V \Longrightarrow U_x \subseteq f^{-1} (V)$

Therefore for all $ \displaystyle x \in f^{-1} (V)$ we have $ \displaystyle x \in U_x \subseteq f^{-1} (V)$

Therefore $ \displaystyle f^{-1} (V)$ is open by Proposition 7.2 ...

Therefore $ \displaystyle f^{-1} (V) \in T_X$ ... ...

Hope someone can help ... ...

Peter

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The above post mentions Proposition 7.2 so I am providing text of the same together with the start of Chapter 7 in order to provide necessary context, definitions and notation ... as follows ... ...

Sutherland - 1 - Defn 7.1 and Propn 7.2 ... PART 1 ... .png

Sutherland - 2 - Defn 7.1 and Propn 7.2 ... PART 2 ... .png

Hope that helps ...

Peter