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how to find the limit point?

order topology and discrete topologySubspace topology and order topologySubspace topology and order topology (2)Convexity implies the equivalence of order and subspace topologiesOrder topology neighborhoodOpen set in order topologyquestions about subspace topology, order topology and convex subsetIs the interval $((a,b), (a,d))$ is open in $mathbbRtimes mathbbR$ which is equipped with the dictionary order topologyMunkres Example 16.3What are the limit points of $X times mathbbN$?

0

$begingroup$

Let $X=0,1,2 $ with natural order topology and give $mathbbN$ its natural order topology. Consider $X times mathbbN$ with the dictionary order topology.

What are the limit points of $X times mathbbN$ ?

Answer:

$X times mathbbN=(0,1), (0,2), cdots, (0, infty), (1,1),(1,2), cdots, (1, infty), (2,1), (2,2), cdots, (2, infty) , cdots $

But how to find the limit point?

If I know the basic open sets in $X times mathbbN$ then I can find the limits poinst.

Since order topology on $mathbbN$ is discrete topology and same for $X$, so the dictionary order topology on $X times mathbbN$ means what?

I think any neighborhood of any point in $X times mathbbN$ contains points of $X times mathbbN$ and so all its points are limits points.

help me

order-topology

share|cite|improve this question

edited Mar 21 at 18:32

M. A. SARKAR

asked Mar 21 at 17:56

M. A. SARKARM. A. SARKAR

2,4621820

$endgroup$

add a comment | 

0

$begingroup$

Let $X=0,1,2 $ with natural order topology and give $mathbbN$ its natural order topology. Consider $X times mathbbN$ with the dictionary order topology.

What are the limit points of $X times mathbbN$ ?

Answer:

$X times mathbbN=(0,1), (0,2), cdots, (0, infty), (1,1),(1,2), cdots, (1, infty), (2,1), (2,2), cdots, (2, infty) , cdots $

But how to find the limit point?

If I know the basic open sets in $X times mathbbN$ then I can find the limits poinst.

Since order topology on $mathbbN$ is discrete topology and same for $X$, so the dictionary order topology on $X times mathbbN$ means what?

I think any neighborhood of any point in $X times mathbbN$ contains points of $X times mathbbN$ and so all its points are limits points.

help me

order-topology

share|cite|improve this question

edited Mar 21 at 18:32

M. A. SARKAR

asked Mar 21 at 17:56

M. A. SARKARM. A. SARKAR

2,4621820

$endgroup$

add a comment | 

$begingroup$

Let $X=0,1,2 $ with natural order topology and give $mathbbN$ its natural order topology. Consider $X times mathbbN$ with the dictionary order topology.

What are the limit points of $X times mathbbN$ ?

Answer:

$X times mathbbN=(0,1), (0,2), cdots, (0, infty), (1,1),(1,2), cdots, (1, infty), (2,1), (2,2), cdots, (2, infty) , cdots $

But how to find the limit point?

If I know the basic open sets in $X times mathbbN$ then I can find the limits poinst.

Since order topology on $mathbbN$ is discrete topology and same for $X$, so the dictionary order topology on $X times mathbbN$ means what?

I think any neighborhood of any point in $X times mathbbN$ contains points of $X times mathbbN$ and so all its points are limits points.

help me

order-topology

share|cite|improve this question

edited Mar 21 at 18:32

M. A. SARKAR

asked Mar 21 at 17:56

M. A. SARKARM. A. SARKAR

2,4621820

$endgroup$

share|cite|improve this question

edited Mar 21 at 18:32

M. A. SARKAR

asked Mar 21 at 17:56

M. A. SARKARM. A. SARKAR

2,4621820

share|cite|improve this question

edited Mar 21 at 18:32

M. A. SARKAR

asked Mar 21 at 17:56

M. A. SARKARM. A. SARKAR

2,4621820

asked Mar 21 at 17:56

M. A. SARKARM. A. SARKAR

2,4621820

asked Mar 21 at 17:56

M. A. SARKARM. A. SARKAR

2,4621820