Union of connected subsets is connected if intersection is nonempty












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Let $mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $bigcapmathscr{F}neemptyset$. Prove that $bigcupmathscr{F}$ is connected.




If $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty subsets $A,B$. Let $x$ be a point in $bigcapmathscr{F}$. Then either $xin A$ or $xin B$. I don't know where to go from here.










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    20















    Let $mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $bigcapmathscr{F}neemptyset$. Prove that $bigcupmathscr{F}$ is connected.




    If $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty subsets $A,B$. Let $x$ be a point in $bigcapmathscr{F}$. Then either $xin A$ or $xin B$. I don't know where to go from here.










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      20












      20








      20


      9






      Let $mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $bigcapmathscr{F}neemptyset$. Prove that $bigcupmathscr{F}$ is connected.




      If $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty subsets $A,B$. Let $x$ be a point in $bigcapmathscr{F}$. Then either $xin A$ or $xin B$. I don't know where to go from here.










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      Let $mathscr{F}$ be a collection of connected subsets of a metric space $M$ such that $bigcapmathscr{F}neemptyset$. Prove that $bigcupmathscr{F}$ is connected.




      If $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty subsets $A,B$. Let $x$ be a point in $bigcapmathscr{F}$. Then either $xin A$ or $xin B$. I don't know where to go from here.







      general-topology connectedness






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      edited Mar 22 '17 at 5:57









      Martin Sleziak

      44.7k7115271




      44.7k7115271










      asked Jun 20 '13 at 0:39









      PJ Miller

      2,57322769




      2,57322769






















          5 Answers
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          27














          HINT: You’re actually about halfway there, though you omitted an important qualification of $A$ and $B$: if $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty, relatively open subsets $A$ and $B$. Now fix $xinbigcapmathscr{F}$, and without loss of generality assume that $xin A$. $Bnevarnothing$, so pick any $yin B$. Then there is some $Finmathscr{F}$ such that $yin F$, and of course $xin F$. Thus, $xin Acap F$, and $yin Bcap F$, so $Acap Fnevarnothingne Bcap F$. Why is this a contradiction?






          share|cite|improve this answer





























            10














            Use that $X$ is connected if and only if the only continuous functions $f:Xto{0,1}$ are constant, where ${0,1}$ is endowed with the discrete topology.



            Now, you know each $F$ in $mathscr F$ is connected. Consider $f:bigcup mathscr Fto{0,1}$, $f$ continuous.



            Take $alpha inbigcapmathscr F$. Look at $f(alpha)$, and at $fmid_{F}:bigcup mathscr Fto{0,1}$ for any $Finmathscr F$.





            Since you mention metric spaces, I am not sure if you know about the first thing I mention, so let's prove it:




            THM Let $(X,mathscr T)$ be a metric (or more generally, a topological) space. Then $X$ is connected if and only if whenever $f:Xto{0,1}$ is continuous, it is constant. The space ${0,1}$ is endowed with the discrete metric (topology), that is, the open sets are $varnothing,{0},{1},{0,1}$.




            P First, suppose $X$ is disconnected, say by $A,B$, so $Acup B=X$ and $Acap B=varnothing$, $A,B$ open. Define $f:Xto{0,1}$ by $$f(x)=begin{cases}1& ; ; xin A\0&; ; xin Bend{cases}$$



            Then $f$ is continuous because $f^{-1}(G)$ is open for any open $G$ in ${0,1}$ (this is simply a case by case verification), yet it is not constant. Now suppose $f:Xto{0,1}$ is continuous but not constant. Set $A={x:f(x)=1}=f^{-1}({1})$ and $B={x:f(x)=0}=f^{-1}({0})$. By hypothesis, $A,Bneq varnothing$. Morover, both are open, since they are the preimage of open sets under a continuous map, and $Acup B=X$ and $Acap B=varnothing$. Thus $X$ is disconnected. $blacktriangle$






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              4














              I like "clopen" sets... it is a set that is at the same time closed and open. A topological space $X$ is not connected if, and only if, for any point $a in X$, you have a "non-trivial" clopen set $C$ containing $a$. That is, a clopen set such that $emptyset neq C subsetneq X$.



              Take $a in bigcap mathscr{F}$. Then, take a clopen set $C subset bigcup mathscr{F}$ containing $a$. In the relative topology, for any $X in mathscr{F}$, $C cap X$ is a clopen set. Since $X$ is connected, $C cap X = X$. That is, $X subset C$ for every $X in mathscr{F}$. Therefore, $bigcup mathscr{F} subset C$.



              Therefore, $bigcup mathscr{F}$ has no "non-trivial" clopen set, and so, it is connected.





              The proof admits some variants. For example, if you take any non-empty clopen set $C'$, it will intersect some $X in mathscr{F}$. From connectedness, we have that $X subset C'$. In particular, $a in C'$.






              share|cite|improve this answer































                3














                The quickest way is using functions from $bigcupmathcal{F}$ to ${0,1}$, However you can do it directly.



                Let $A, B$ be a partition of $bigcupmathcal{F}$ into open, disjoint subsets. WLOG assume that $A$ is non-empty. Then there is some $xin A$ and hence some $Fin mathcal{F}$ so that $xin F$. As $F$ is connected it follows that $Fsubseteq A$, hence $bigcap mathcal{F} subseteq F subseteq A$. As this intersection is non-empty every $Gin mathcal{F}$ has a point in $A$ and by the connectness of each $G$ then $Gsubseteq A$. Hence every member of $mathcal{F}$ is a subset of $A$ and so $B$ is empty.






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                  0














                  In the before last post, with the "clopen" sets, I believe there is a detail missing: it is not immediately clear (but true) that if $C$ is a clopen set with respect to the subset topology on $bigcupmathscr{F}$, that then $Ccap X$ is a clopen set in $X$, $Xinmathscr{F}$, with respect to the original topology, which is what you need.






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                    5 Answers
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                    27














                    HINT: You’re actually about halfway there, though you omitted an important qualification of $A$ and $B$: if $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty, relatively open subsets $A$ and $B$. Now fix $xinbigcapmathscr{F}$, and without loss of generality assume that $xin A$. $Bnevarnothing$, so pick any $yin B$. Then there is some $Finmathscr{F}$ such that $yin F$, and of course $xin F$. Thus, $xin Acap F$, and $yin Bcap F$, so $Acap Fnevarnothingne Bcap F$. Why is this a contradiction?






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                      27














                      HINT: You’re actually about halfway there, though you omitted an important qualification of $A$ and $B$: if $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty, relatively open subsets $A$ and $B$. Now fix $xinbigcapmathscr{F}$, and without loss of generality assume that $xin A$. $Bnevarnothing$, so pick any $yin B$. Then there is some $Finmathscr{F}$ such that $yin F$, and of course $xin F$. Thus, $xin Acap F$, and $yin Bcap F$, so $Acap Fnevarnothingne Bcap F$. Why is this a contradiction?






                      share|cite|improve this answer
























                        27












                        27








                        27






                        HINT: You’re actually about halfway there, though you omitted an important qualification of $A$ and $B$: if $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty, relatively open subsets $A$ and $B$. Now fix $xinbigcapmathscr{F}$, and without loss of generality assume that $xin A$. $Bnevarnothing$, so pick any $yin B$. Then there is some $Finmathscr{F}$ such that $yin F$, and of course $xin F$. Thus, $xin Acap F$, and $yin Bcap F$, so $Acap Fnevarnothingne Bcap F$. Why is this a contradiction?






                        share|cite|improve this answer












                        HINT: You’re actually about halfway there, though you omitted an important qualification of $A$ and $B$: if $bigcupmathscr{F}$ is not connected, then it can be partitioned into two disjoint, non-empty, relatively open subsets $A$ and $B$. Now fix $xinbigcapmathscr{F}$, and without loss of generality assume that $xin A$. $Bnevarnothing$, so pick any $yin B$. Then there is some $Finmathscr{F}$ such that $yin F$, and of course $xin F$. Thus, $xin Acap F$, and $yin Bcap F$, so $Acap Fnevarnothingne Bcap F$. Why is this a contradiction?







                        share|cite|improve this answer












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                        answered Jun 20 '13 at 0:46









                        Brian M. Scott

                        455k38505908




                        455k38505908























                            10














                            Use that $X$ is connected if and only if the only continuous functions $f:Xto{0,1}$ are constant, where ${0,1}$ is endowed with the discrete topology.



                            Now, you know each $F$ in $mathscr F$ is connected. Consider $f:bigcup mathscr Fto{0,1}$, $f$ continuous.



                            Take $alpha inbigcapmathscr F$. Look at $f(alpha)$, and at $fmid_{F}:bigcup mathscr Fto{0,1}$ for any $Finmathscr F$.





                            Since you mention metric spaces, I am not sure if you know about the first thing I mention, so let's prove it:




                            THM Let $(X,mathscr T)$ be a metric (or more generally, a topological) space. Then $X$ is connected if and only if whenever $f:Xto{0,1}$ is continuous, it is constant. The space ${0,1}$ is endowed with the discrete metric (topology), that is, the open sets are $varnothing,{0},{1},{0,1}$.




                            P First, suppose $X$ is disconnected, say by $A,B$, so $Acup B=X$ and $Acap B=varnothing$, $A,B$ open. Define $f:Xto{0,1}$ by $$f(x)=begin{cases}1& ; ; xin A\0&; ; xin Bend{cases}$$



                            Then $f$ is continuous because $f^{-1}(G)$ is open for any open $G$ in ${0,1}$ (this is simply a case by case verification), yet it is not constant. Now suppose $f:Xto{0,1}$ is continuous but not constant. Set $A={x:f(x)=1}=f^{-1}({1})$ and $B={x:f(x)=0}=f^{-1}({0})$. By hypothesis, $A,Bneq varnothing$. Morover, both are open, since they are the preimage of open sets under a continuous map, and $Acup B=X$ and $Acap B=varnothing$. Thus $X$ is disconnected. $blacktriangle$






                            share|cite|improve this answer




























                              10














                              Use that $X$ is connected if and only if the only continuous functions $f:Xto{0,1}$ are constant, where ${0,1}$ is endowed with the discrete topology.



                              Now, you know each $F$ in $mathscr F$ is connected. Consider $f:bigcup mathscr Fto{0,1}$, $f$ continuous.



                              Take $alpha inbigcapmathscr F$. Look at $f(alpha)$, and at $fmid_{F}:bigcup mathscr Fto{0,1}$ for any $Finmathscr F$.





                              Since you mention metric spaces, I am not sure if you know about the first thing I mention, so let's prove it:




                              THM Let $(X,mathscr T)$ be a metric (or more generally, a topological) space. Then $X$ is connected if and only if whenever $f:Xto{0,1}$ is continuous, it is constant. The space ${0,1}$ is endowed with the discrete metric (topology), that is, the open sets are $varnothing,{0},{1},{0,1}$.




                              P First, suppose $X$ is disconnected, say by $A,B$, so $Acup B=X$ and $Acap B=varnothing$, $A,B$ open. Define $f:Xto{0,1}$ by $$f(x)=begin{cases}1& ; ; xin A\0&; ; xin Bend{cases}$$



                              Then $f$ is continuous because $f^{-1}(G)$ is open for any open $G$ in ${0,1}$ (this is simply a case by case verification), yet it is not constant. Now suppose $f:Xto{0,1}$ is continuous but not constant. Set $A={x:f(x)=1}=f^{-1}({1})$ and $B={x:f(x)=0}=f^{-1}({0})$. By hypothesis, $A,Bneq varnothing$. Morover, both are open, since they are the preimage of open sets under a continuous map, and $Acup B=X$ and $Acap B=varnothing$. Thus $X$ is disconnected. $blacktriangle$






                              share|cite|improve this answer


























                                10












                                10








                                10






                                Use that $X$ is connected if and only if the only continuous functions $f:Xto{0,1}$ are constant, where ${0,1}$ is endowed with the discrete topology.



                                Now, you know each $F$ in $mathscr F$ is connected. Consider $f:bigcup mathscr Fto{0,1}$, $f$ continuous.



                                Take $alpha inbigcapmathscr F$. Look at $f(alpha)$, and at $fmid_{F}:bigcup mathscr Fto{0,1}$ for any $Finmathscr F$.





                                Since you mention metric spaces, I am not sure if you know about the first thing I mention, so let's prove it:




                                THM Let $(X,mathscr T)$ be a metric (or more generally, a topological) space. Then $X$ is connected if and only if whenever $f:Xto{0,1}$ is continuous, it is constant. The space ${0,1}$ is endowed with the discrete metric (topology), that is, the open sets are $varnothing,{0},{1},{0,1}$.




                                P First, suppose $X$ is disconnected, say by $A,B$, so $Acup B=X$ and $Acap B=varnothing$, $A,B$ open. Define $f:Xto{0,1}$ by $$f(x)=begin{cases}1& ; ; xin A\0&; ; xin Bend{cases}$$



                                Then $f$ is continuous because $f^{-1}(G)$ is open for any open $G$ in ${0,1}$ (this is simply a case by case verification), yet it is not constant. Now suppose $f:Xto{0,1}$ is continuous but not constant. Set $A={x:f(x)=1}=f^{-1}({1})$ and $B={x:f(x)=0}=f^{-1}({0})$. By hypothesis, $A,Bneq varnothing$. Morover, both are open, since they are the preimage of open sets under a continuous map, and $Acup B=X$ and $Acap B=varnothing$. Thus $X$ is disconnected. $blacktriangle$






                                share|cite|improve this answer














                                Use that $X$ is connected if and only if the only continuous functions $f:Xto{0,1}$ are constant, where ${0,1}$ is endowed with the discrete topology.



                                Now, you know each $F$ in $mathscr F$ is connected. Consider $f:bigcup mathscr Fto{0,1}$, $f$ continuous.



                                Take $alpha inbigcapmathscr F$. Look at $f(alpha)$, and at $fmid_{F}:bigcup mathscr Fto{0,1}$ for any $Finmathscr F$.





                                Since you mention metric spaces, I am not sure if you know about the first thing I mention, so let's prove it:




                                THM Let $(X,mathscr T)$ be a metric (or more generally, a topological) space. Then $X$ is connected if and only if whenever $f:Xto{0,1}$ is continuous, it is constant. The space ${0,1}$ is endowed with the discrete metric (topology), that is, the open sets are $varnothing,{0},{1},{0,1}$.




                                P First, suppose $X$ is disconnected, say by $A,B$, so $Acup B=X$ and $Acap B=varnothing$, $A,B$ open. Define $f:Xto{0,1}$ by $$f(x)=begin{cases}1& ; ; xin A\0&; ; xin Bend{cases}$$



                                Then $f$ is continuous because $f^{-1}(G)$ is open for any open $G$ in ${0,1}$ (this is simply a case by case verification), yet it is not constant. Now suppose $f:Xto{0,1}$ is continuous but not constant. Set $A={x:f(x)=1}=f^{-1}({1})$ and $B={x:f(x)=0}=f^{-1}({0})$. By hypothesis, $A,Bneq varnothing$. Morover, both are open, since they are the preimage of open sets under a continuous map, and $Acup B=X$ and $Acap B=varnothing$. Thus $X$ is disconnected. $blacktriangle$







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                                edited Jun 2 '16 at 9:48

























                                answered Jun 20 '13 at 0:42









                                Pedro Tamaroff

                                96.3k10151296




                                96.3k10151296























                                    4














                                    I like "clopen" sets... it is a set that is at the same time closed and open. A topological space $X$ is not connected if, and only if, for any point $a in X$, you have a "non-trivial" clopen set $C$ containing $a$. That is, a clopen set such that $emptyset neq C subsetneq X$.



                                    Take $a in bigcap mathscr{F}$. Then, take a clopen set $C subset bigcup mathscr{F}$ containing $a$. In the relative topology, for any $X in mathscr{F}$, $C cap X$ is a clopen set. Since $X$ is connected, $C cap X = X$. That is, $X subset C$ for every $X in mathscr{F}$. Therefore, $bigcup mathscr{F} subset C$.



                                    Therefore, $bigcup mathscr{F}$ has no "non-trivial" clopen set, and so, it is connected.





                                    The proof admits some variants. For example, if you take any non-empty clopen set $C'$, it will intersect some $X in mathscr{F}$. From connectedness, we have that $X subset C'$. In particular, $a in C'$.






                                    share|cite|improve this answer




























                                      4














                                      I like "clopen" sets... it is a set that is at the same time closed and open. A topological space $X$ is not connected if, and only if, for any point $a in X$, you have a "non-trivial" clopen set $C$ containing $a$. That is, a clopen set such that $emptyset neq C subsetneq X$.



                                      Take $a in bigcap mathscr{F}$. Then, take a clopen set $C subset bigcup mathscr{F}$ containing $a$. In the relative topology, for any $X in mathscr{F}$, $C cap X$ is a clopen set. Since $X$ is connected, $C cap X = X$. That is, $X subset C$ for every $X in mathscr{F}$. Therefore, $bigcup mathscr{F} subset C$.



                                      Therefore, $bigcup mathscr{F}$ has no "non-trivial" clopen set, and so, it is connected.





                                      The proof admits some variants. For example, if you take any non-empty clopen set $C'$, it will intersect some $X in mathscr{F}$. From connectedness, we have that $X subset C'$. In particular, $a in C'$.






                                      share|cite|improve this answer


























                                        4












                                        4








                                        4






                                        I like "clopen" sets... it is a set that is at the same time closed and open. A topological space $X$ is not connected if, and only if, for any point $a in X$, you have a "non-trivial" clopen set $C$ containing $a$. That is, a clopen set such that $emptyset neq C subsetneq X$.



                                        Take $a in bigcap mathscr{F}$. Then, take a clopen set $C subset bigcup mathscr{F}$ containing $a$. In the relative topology, for any $X in mathscr{F}$, $C cap X$ is a clopen set. Since $X$ is connected, $C cap X = X$. That is, $X subset C$ for every $X in mathscr{F}$. Therefore, $bigcup mathscr{F} subset C$.



                                        Therefore, $bigcup mathscr{F}$ has no "non-trivial" clopen set, and so, it is connected.





                                        The proof admits some variants. For example, if you take any non-empty clopen set $C'$, it will intersect some $X in mathscr{F}$. From connectedness, we have that $X subset C'$. In particular, $a in C'$.






                                        share|cite|improve this answer














                                        I like "clopen" sets... it is a set that is at the same time closed and open. A topological space $X$ is not connected if, and only if, for any point $a in X$, you have a "non-trivial" clopen set $C$ containing $a$. That is, a clopen set such that $emptyset neq C subsetneq X$.



                                        Take $a in bigcap mathscr{F}$. Then, take a clopen set $C subset bigcup mathscr{F}$ containing $a$. In the relative topology, for any $X in mathscr{F}$, $C cap X$ is a clopen set. Since $X$ is connected, $C cap X = X$. That is, $X subset C$ for every $X in mathscr{F}$. Therefore, $bigcup mathscr{F} subset C$.



                                        Therefore, $bigcup mathscr{F}$ has no "non-trivial" clopen set, and so, it is connected.





                                        The proof admits some variants. For example, if you take any non-empty clopen set $C'$, it will intersect some $X in mathscr{F}$. From connectedness, we have that $X subset C'$. In particular, $a in C'$.







                                        share|cite|improve this answer














                                        share|cite|improve this answer



                                        share|cite|improve this answer








                                        edited Jun 20 '13 at 1:51

























                                        answered Jun 20 '13 at 1:38









                                        André Caldas

                                        3,3971227




                                        3,3971227























                                            3














                                            The quickest way is using functions from $bigcupmathcal{F}$ to ${0,1}$, However you can do it directly.



                                            Let $A, B$ be a partition of $bigcupmathcal{F}$ into open, disjoint subsets. WLOG assume that $A$ is non-empty. Then there is some $xin A$ and hence some $Fin mathcal{F}$ so that $xin F$. As $F$ is connected it follows that $Fsubseteq A$, hence $bigcap mathcal{F} subseteq F subseteq A$. As this intersection is non-empty every $Gin mathcal{F}$ has a point in $A$ and by the connectness of each $G$ then $Gsubseteq A$. Hence every member of $mathcal{F}$ is a subset of $A$ and so $B$ is empty.






                                            share|cite|improve this answer


























                                              3














                                              The quickest way is using functions from $bigcupmathcal{F}$ to ${0,1}$, However you can do it directly.



                                              Let $A, B$ be a partition of $bigcupmathcal{F}$ into open, disjoint subsets. WLOG assume that $A$ is non-empty. Then there is some $xin A$ and hence some $Fin mathcal{F}$ so that $xin F$. As $F$ is connected it follows that $Fsubseteq A$, hence $bigcap mathcal{F} subseteq F subseteq A$. As this intersection is non-empty every $Gin mathcal{F}$ has a point in $A$ and by the connectness of each $G$ then $Gsubseteq A$. Hence every member of $mathcal{F}$ is a subset of $A$ and so $B$ is empty.






                                              share|cite|improve this answer
























                                                3












                                                3








                                                3






                                                The quickest way is using functions from $bigcupmathcal{F}$ to ${0,1}$, However you can do it directly.



                                                Let $A, B$ be a partition of $bigcupmathcal{F}$ into open, disjoint subsets. WLOG assume that $A$ is non-empty. Then there is some $xin A$ and hence some $Fin mathcal{F}$ so that $xin F$. As $F$ is connected it follows that $Fsubseteq A$, hence $bigcap mathcal{F} subseteq F subseteq A$. As this intersection is non-empty every $Gin mathcal{F}$ has a point in $A$ and by the connectness of each $G$ then $Gsubseteq A$. Hence every member of $mathcal{F}$ is a subset of $A$ and so $B$ is empty.






                                                share|cite|improve this answer












                                                The quickest way is using functions from $bigcupmathcal{F}$ to ${0,1}$, However you can do it directly.



                                                Let $A, B$ be a partition of $bigcupmathcal{F}$ into open, disjoint subsets. WLOG assume that $A$ is non-empty. Then there is some $xin A$ and hence some $Fin mathcal{F}$ so that $xin F$. As $F$ is connected it follows that $Fsubseteq A$, hence $bigcap mathcal{F} subseteq F subseteq A$. As this intersection is non-empty every $Gin mathcal{F}$ has a point in $A$ and by the connectness of each $G$ then $Gsubseteq A$. Hence every member of $mathcal{F}$ is a subset of $A$ and so $B$ is empty.







                                                share|cite|improve this answer












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                                                answered Jun 20 '13 at 0:49









                                                James

                                                4,2401821




                                                4,2401821























                                                    0














                                                    In the before last post, with the "clopen" sets, I believe there is a detail missing: it is not immediately clear (but true) that if $C$ is a clopen set with respect to the subset topology on $bigcupmathscr{F}$, that then $Ccap X$ is a clopen set in $X$, $Xinmathscr{F}$, with respect to the original topology, which is what you need.






                                                    share|cite|improve this answer




























                                                      0














                                                      In the before last post, with the "clopen" sets, I believe there is a detail missing: it is not immediately clear (but true) that if $C$ is a clopen set with respect to the subset topology on $bigcupmathscr{F}$, that then $Ccap X$ is a clopen set in $X$, $Xinmathscr{F}$, with respect to the original topology, which is what you need.






                                                      share|cite|improve this answer


























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                                                        In the before last post, with the "clopen" sets, I believe there is a detail missing: it is not immediately clear (but true) that if $C$ is a clopen set with respect to the subset topology on $bigcupmathscr{F}$, that then $Ccap X$ is a clopen set in $X$, $Xinmathscr{F}$, with respect to the original topology, which is what you need.






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                                                        In the before last post, with the "clopen" sets, I believe there is a detail missing: it is not immediately clear (but true) that if $C$ is a clopen set with respect to the subset topology on $bigcupmathscr{F}$, that then $Ccap X$ is a clopen set in $X$, $Xinmathscr{F}$, with respect to the original topology, which is what you need.







                                                        share|cite|improve this answer














                                                        share|cite|improve this answer



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                                                        edited Apr 15 '17 at 20:06

























                                                        answered Apr 15 '17 at 16:03









                                                        Michiel Vermeulen

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