Is the set ${2,3,5}$ connected?
I read somewhere that a connected set in the digital line topology is a subset of consecutive integers. Recall that the digital line topology is a topology on $mathbb{Z}$ with basis elements ${n}$ for $n$ odd and ${n-1, n, n+1}$ for $n$ even. However, if we consider the set ${2,3,5}$, then this set must be disconnected based on the above statement, as it is missing the integer $4$. In order to show that this set is disconnected, we must show that it has a separation (or that it is the union of two disjoint nonempty open sets). ${5}$ is open in the digital line topology, as $5$ is odd. However, for the case of the set ${2,3}$ it's trickier. We can show that this set is not open, as we cannot construct an open neighborhood around ${2}$ such that the neighborhood is entirely contained in the two-tuple set. How do I show that this set is indeed disconnected?
general-topology connectedness
asked Nov 28 at 22:50
Derek Adams
510413
add a comment |
I read somewhere that a connected set in the digital line topology is a subset of consecutive integers. Recall that the digital line topology is a topology on $mathbb{Z}$ with basis elements ${n}$ for $n$ odd and ${n-1, n, n+1}$ for $n$ even. However, if we consider the set ${2,3,5}$, then this set must be disconnected based on the above statement, as it is missing the integer $4$. In order to show that this set is disconnected, we must show that it has a separation (or that it is the union of two disjoint nonempty open sets). ${5}$ is open in the digital line topology, as $5$ is odd. However, for the case of the set ${2,3}$ it's trickier. We can show that this set is not open, as we cannot construct an open neighborhood around ${2}$ such that the neighborhood is entirely contained in the two-tuple set. How do I show that this set is indeed disconnected?
general-topology connectedness
asked Nov 28 at 22:50
Derek Adams
510413
add a comment |
I read somewhere that a connected set in the digital line topology is a subset of consecutive integers. Recall that the digital line topology is a topology on $mathbb{Z}$ with basis elements ${n}$ for $n$ odd and ${n-1, n, n+1}$ for $n$ even. However, if we consider the set ${2,3,5}$, then this set must be disconnected based on the above statement, as it is missing the integer $4$. In order to show that this set is disconnected, we must show that it has a separation (or that it is the union of two disjoint nonempty open sets). ${5}$ is open in the digital line topology, as $5$ is odd. However, for the case of the set ${2,3}$ it's trickier. We can show that this set is not open, as we cannot construct an open neighborhood around ${2}$ such that the neighborhood is entirely contained in the two-tuple set. How do I show that this set is indeed disconnected?
general-topology connectedness
asked Nov 28 at 22:50
Derek Adams
510413
I read somewhere that a connected set in the digital line topology is a subset of consecutive integers. Recall that the digital line topology is a topology on $mathbb{Z}$ with basis elements ${n}$ for $n$ odd and ${n-1, n, n+1}$ for $n$ even. However, if we consider the set ${2,3,5}$, then this set must be disconnected based on the above statement, as it is missing the integer $4$. In order to show that this set is disconnected, we must show that it has a separation (or that it is the union of two disjoint nonempty open sets). ${5}$ is open in the digital line topology, as $5$ is odd. However, for the case of the set ${2,3}$ it's trickier. We can show that this set is not open, as we cannot construct an open neighborhood around ${2}$ such that the neighborhood is entirely contained in the two-tuple set. How do I show that this set is indeed disconnected?
general-topology connectedness
general-topology connectedness
asked Nov 28 at 22:50
Derek Adams
510413
asked Nov 28 at 22:50
Derek Adams
510413
asked Nov 28 at 22:50
Derek Adams
510413
asked Nov 28 at 22:50
Derek Adams
510413
asked Nov 28 at 22:50
Derek Adams
510413
510413
add a comment |
add a comment |
2 Answers
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The set ${2,3}$ is open, since it is equal to ${1,2,3}cap{2,3,5}$ and ${1,2,3}$ is open set in $mathbb Z$.
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
Such a simple oversight. Thank you.
– Derek Adams
Nov 28 at 22:58
add a comment |
To say that a subset is disconnected is to say that it is disconnected under the subspace topology. To show that ${2,3,5}$ is disconnected, you can show that it is a disjoint union of sets that are open under the subspace topology. Remember that a set $S$ is open under the subspace topology here if $S=Tcap{2,3,5}$ where $T$ is open in the original topology, in this case, the digital line topology. Then ${2,3,5}={2,3}cup {5}$, the former of which is open because ${2,3}={1,2,3}cap{2,3,5}$, and ${1,2,3}$ is open in the digital line topology, and the latter is open as you pointed out.
answered Nov 28 at 22:58
Kevin Long
3,36621330
add a comment |
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2 Answers
2
active
oldest
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2 Answers
2
active
oldest
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active
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active
oldest
votes
The set ${2,3}$ is open, since it is equal to ${1,2,3}cap{2,3,5}$ and ${1,2,3}$ is open set in $mathbb Z$.
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
Such a simple oversight. Thank you.
– Derek Adams
Nov 28 at 22:58
add a comment |
The set ${2,3}$ is open, since it is equal to ${1,2,3}cap{2,3,5}$ and ${1,2,3}$ is open set in $mathbb Z$.
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
Such a simple oversight. Thank you.
– Derek Adams
Nov 28 at 22:58
add a comment |
The set ${2,3}$ is open, since it is equal to ${1,2,3}cap{2,3,5}$ and ${1,2,3}$ is open set in $mathbb Z$.
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
The set ${2,3}$ is open, since it is equal to ${1,2,3}cap{2,3,5}$ and ${1,2,3}$ is open set in $mathbb Z$.
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
answered Nov 28 at 22:57
José Carlos Santos
148k22117218
148k22117218
Such a simple oversight. Thank you.
– Derek Adams
Nov 28 at 22:58
add a comment |
Such a simple oversight. Thank you.
– Derek Adams
Nov 28 at 22:58
Such a simple oversight. Thank you.
– Derek Adams
Nov 28 at 22:58
Such a simple oversight. Thank you.
– Derek Adams
Nov 28 at 22:58
add a comment |
To say that a subset is disconnected is to say that it is disconnected under the subspace topology. To show that ${2,3,5}$ is disconnected, you can show that it is a disjoint union of sets that are open under the subspace topology. Remember that a set $S$ is open under the subspace topology here if $S=Tcap{2,3,5}$ where $T$ is open in the original topology, in this case, the digital line topology. Then ${2,3,5}={2,3}cup {5}$, the former of which is open because ${2,3}={1,2,3}cap{2,3,5}$, and ${1,2,3}$ is open in the digital line topology, and the latter is open as you pointed out.
answered Nov 28 at 22:58
Kevin Long
3,36621330
add a comment |
To say that a subset is disconnected is to say that it is disconnected under the subspace topology. To show that ${2,3,5}$ is disconnected, you can show that it is a disjoint union of sets that are open under the subspace topology. Remember that a set $S$ is open under the subspace topology here if $S=Tcap{2,3,5}$ where $T$ is open in the original topology, in this case, the digital line topology. Then ${2,3,5}={2,3}cup {5}$, the former of which is open because ${2,3}={1,2,3}cap{2,3,5}$, and ${1,2,3}$ is open in the digital line topology, and the latter is open as you pointed out.
answered Nov 28 at 22:58
Kevin Long
3,36621330
add a comment |
To say that a subset is disconnected is to say that it is disconnected under the subspace topology. To show that ${2,3,5}$ is disconnected, you can show that it is a disjoint union of sets that are open under the subspace topology. Remember that a set $S$ is open under the subspace topology here if $S=Tcap{2,3,5}$ where $T$ is open in the original topology, in this case, the digital line topology. Then ${2,3,5}={2,3}cup {5}$, the former of which is open because ${2,3}={1,2,3}cap{2,3,5}$, and ${1,2,3}$ is open in the digital line topology, and the latter is open as you pointed out.
answered Nov 28 at 22:58
Kevin Long
3,36621330
To say that a subset is disconnected is to say that it is disconnected under the subspace topology. To show that ${2,3,5}$ is disconnected, you can show that it is a disjoint union of sets that are open under the subspace topology. Remember that a set $S$ is open under the subspace topology here if $S=Tcap{2,3,5}$ where $T$ is open in the original topology, in this case, the digital line topology. Then ${2,3,5}={2,3}cup {5}$, the former of which is open because ${2,3}={1,2,3}cap{2,3,5}$, and ${1,2,3}$ is open in the digital line topology, and the latter is open as you pointed out.
answered Nov 28 at 22:58
Kevin Long
3,36621330
answered Nov 28 at 22:58
Kevin Long
3,36621330
answered Nov 28 at 22:58
Kevin Long
3,36621330
answered Nov 28 at 22:58
Kevin Long
3,36621330
3,36621330
add a comment |
add a comment |
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