What's the definition of coherent topology in Munkres Topology?











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Munkres Topology. In Section 71, coherent is "if" but on Wikipedia (Coherent topology), it's "if and only if"



enter image description here



This can be seen in Example 1 of Section 83



enter image description here



We suppose $D cap A_{alpha}$ is closed in $A_{alpha}$ and then must show $D$ is closed in $X$. I don't think we also suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



But in the definition of subgraph (also in Section 83), coherent is "if and only if" (I am aware of the errata by Barbara and Jim Munkres for this definition but irrelevant I think).



enter image description here



I was expecting to see that we suppose $D cap A_{beta}$ is closed in $A_{beta}$ and then must show $D$ is closed in $Y$, but we actually also show We suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



What's going on?



My guess (I came up with one only after typing it all up)



Definitions are "if and only if". If there is no specified topology for a space $Z$, then coherence is just "if" and then "only if" follows because coherence is the definition. If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology.



I think I figured it out, but I might as well just submit this since I already typed it up.










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    up vote
    0
    down vote

    favorite












    Munkres Topology. In Section 71, coherent is "if" but on Wikipedia (Coherent topology), it's "if and only if"



    enter image description here



    This can be seen in Example 1 of Section 83



    enter image description here



    We suppose $D cap A_{alpha}$ is closed in $A_{alpha}$ and then must show $D$ is closed in $X$. I don't think we also suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



    But in the definition of subgraph (also in Section 83), coherent is "if and only if" (I am aware of the errata by Barbara and Jim Munkres for this definition but irrelevant I think).



    enter image description here



    I was expecting to see that we suppose $D cap A_{beta}$ is closed in $A_{beta}$ and then must show $D$ is closed in $Y$, but we actually also show We suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



    What's going on?



    My guess (I came up with one only after typing it all up)



    Definitions are "if and only if". If there is no specified topology for a space $Z$, then coherence is just "if" and then "only if" follows because coherence is the definition. If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology.



    I think I figured it out, but I might as well just submit this since I already typed it up.










    share|cite|improve this question
























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Munkres Topology. In Section 71, coherent is "if" but on Wikipedia (Coherent topology), it's "if and only if"



      enter image description here



      This can be seen in Example 1 of Section 83



      enter image description here



      We suppose $D cap A_{alpha}$ is closed in $A_{alpha}$ and then must show $D$ is closed in $X$. I don't think we also suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



      But in the definition of subgraph (also in Section 83), coherent is "if and only if" (I am aware of the errata by Barbara and Jim Munkres for this definition but irrelevant I think).



      enter image description here



      I was expecting to see that we suppose $D cap A_{beta}$ is closed in $A_{beta}$ and then must show $D$ is closed in $Y$, but we actually also show We suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



      What's going on?



      My guess (I came up with one only after typing it all up)



      Definitions are "if and only if". If there is no specified topology for a space $Z$, then coherence is just "if" and then "only if" follows because coherence is the definition. If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology.



      I think I figured it out, but I might as well just submit this since I already typed it up.










      share|cite|improve this question













      Munkres Topology. In Section 71, coherent is "if" but on Wikipedia (Coherent topology), it's "if and only if"



      enter image description here



      This can be seen in Example 1 of Section 83



      enter image description here



      We suppose $D cap A_{alpha}$ is closed in $A_{alpha}$ and then must show $D$ is closed in $X$. I don't think we also suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



      But in the definition of subgraph (also in Section 83), coherent is "if and only if" (I am aware of the errata by Barbara and Jim Munkres for this definition but irrelevant I think).



      enter image description here



      I was expecting to see that we suppose $D cap A_{beta}$ is closed in $A_{beta}$ and then must show $D$ is closed in $Y$, but we actually also show We suppose $D$ is closed in $X$ and then show that the $D cap A_{alpha}$'s are closed.



      What's going on?



      My guess (I came up with one only after typing it all up)



      Definitions are "if and only if". If there is no specified topology for a space $Z$, then coherence is just "if" and then "only if" follows because coherence is the definition. If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology.



      I think I figured it out, but I might as well just submit this since I already typed it up.







      abstract-algebra general-topology graph-theory algebraic-topology covering-spaces






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      asked yesterday









      Jack Bauer

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          Definitions, by their nature, are "if and only if". This leads to some authors being lazy when writing them, and they only say "if".



          It's a common enough occurrence and something one should always be aware of. The alternative definition supplied by Munkres definitely seems to be one such case.






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          • Is this correct? "If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology."
            – Jack Bauer
            yesterday











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          up vote
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          Definitions, by their nature, are "if and only if". This leads to some authors being lazy when writing them, and they only say "if".



          It's a common enough occurrence and something one should always be aware of. The alternative definition supplied by Munkres definitely seems to be one such case.






          share|cite|improve this answer





















          • Is this correct? "If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology."
            – Jack Bauer
            yesterday















          up vote
          0
          down vote













          Definitions, by their nature, are "if and only if". This leads to some authors being lazy when writing them, and they only say "if".



          It's a common enough occurrence and something one should always be aware of. The alternative definition supplied by Munkres definitely seems to be one such case.






          share|cite|improve this answer





















          • Is this correct? "If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology."
            – Jack Bauer
            yesterday













          up vote
          0
          down vote










          up vote
          0
          down vote









          Definitions, by their nature, are "if and only if". This leads to some authors being lazy when writing them, and they only say "if".



          It's a common enough occurrence and something one should always be aware of. The alternative definition supplied by Munkres definitely seems to be one such case.






          share|cite|improve this answer












          Definitions, by their nature, are "if and only if". This leads to some authors being lazy when writing them, and they only say "if".



          It's a common enough occurrence and something one should always be aware of. The alternative definition supplied by Munkres definitely seems to be one such case.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered yesterday









          Arthur

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          • Is this correct? "If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology."
            – Jack Bauer
            yesterday


















          • Is this correct? "If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology."
            – Jack Bauer
            yesterday
















          Is this correct? "If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology."
          – Jack Bauer
          yesterday




          Is this correct? "If there is a specified topology on $Z$, such as it having the subspace topology of some other space, then we have to show that coherence, a condition to indicated closedness of sets, and hence openness (of those sets' complements) doesn't conflict with with our new definition of closedness given by the subspace topology."
          – Jack Bauer
          yesterday


















           

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