$X$ is locally compact, Hausdorff, and has a countable family of compact subsets covering $X$, then $X$ is...











up vote
2
down vote

favorite












Suppose $X$ is locally compact, Hausdorff, and has a countable family of compact subsets whose union is all of $X$. I want to show that $X$ is second-countable.
Suppose the family of countable compact subsets are labelled $C_i$, $i in Bbb N$,
Then there are a couple of ideas that come to mind but I am having trouble putting it all together.



1) Taking the complements of the $C_i$ (problem: what if an element is in all of them, also the complements might not satisfy the requirements of a basis)



2) Using local compactness to say that every point has a compact set containing it with an open neighborhood (also containing it), but how will this relate to the $C_i$ ?



Any hints appreciated.










share|cite|improve this question




























    up vote
    2
    down vote

    favorite












    Suppose $X$ is locally compact, Hausdorff, and has a countable family of compact subsets whose union is all of $X$. I want to show that $X$ is second-countable.
    Suppose the family of countable compact subsets are labelled $C_i$, $i in Bbb N$,
    Then there are a couple of ideas that come to mind but I am having trouble putting it all together.



    1) Taking the complements of the $C_i$ (problem: what if an element is in all of them, also the complements might not satisfy the requirements of a basis)



    2) Using local compactness to say that every point has a compact set containing it with an open neighborhood (also containing it), but how will this relate to the $C_i$ ?



    Any hints appreciated.










    share|cite|improve this question


























      up vote
      2
      down vote

      favorite









      up vote
      2
      down vote

      favorite











      Suppose $X$ is locally compact, Hausdorff, and has a countable family of compact subsets whose union is all of $X$. I want to show that $X$ is second-countable.
      Suppose the family of countable compact subsets are labelled $C_i$, $i in Bbb N$,
      Then there are a couple of ideas that come to mind but I am having trouble putting it all together.



      1) Taking the complements of the $C_i$ (problem: what if an element is in all of them, also the complements might not satisfy the requirements of a basis)



      2) Using local compactness to say that every point has a compact set containing it with an open neighborhood (also containing it), but how will this relate to the $C_i$ ?



      Any hints appreciated.










      share|cite|improve this question















      Suppose $X$ is locally compact, Hausdorff, and has a countable family of compact subsets whose union is all of $X$. I want to show that $X$ is second-countable.
      Suppose the family of countable compact subsets are labelled $C_i$, $i in Bbb N$,
      Then there are a couple of ideas that come to mind but I am having trouble putting it all together.



      1) Taking the complements of the $C_i$ (problem: what if an element is in all of them, also the complements might not satisfy the requirements of a basis)



      2) Using local compactness to say that every point has a compact set containing it with an open neighborhood (also containing it), but how will this relate to the $C_i$ ?



      Any hints appreciated.







      general-topology






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Nov 15 at 4:57

























      asked Nov 15 at 4:07









      IntegrateThis

      1,6971717




      1,6971717






















          2 Answers
          2






          active

          oldest

          votes

















          up vote
          3
          down vote



          accepted










          It is not even true that compact Hausdorff spaces are second-countable. (Of course, a compact Hausdorff space is locally compact and can be covered by a countable family of compact subsets.)



          An example is the ordinal space $[ 0 , omega_1 ]$ (where $omega_1$ denotes the least uncountable ordinal).




          • As a linearly-ordered space, $[0 , omega_1 ]$ is Hausdorff.

          • Since $[0 , omega_1]$ has a greatest element, it is compact (this is essentially because there are no infinite strictly decreasing sequences of ordinals).

          • It is not second-countable because any base must include the singletons ${ alpha }$ where $alpha < omega_1$ is a successor ordinal, and there are uncountably many of these.


          More information about this space can be found on the following post on Dan Ma's Topology Blog:




          • The First Uncountable Ordinal


          (If you want a non-compact example of a locally compact σ-compact Hausdorff space that is not second-countable you can just take a disjoint union of countably-infinite copies of $[0,omega_1]$.)






          share|cite|improve this answer























          • So my conclusion is false?
            – IntegrateThis
            Nov 15 at 5:32






          • 1




            @IntegrateThis Correct: it is not true that every locally compact, σ-compact Hausdorff space is second-countable.
            – stochastic randomness
            Nov 15 at 5:55










          • +1............ $[0,omega_1]$ is not even first-countable.
            – DanielWainfleet
            Nov 15 at 7:52










          • Why dooes the base have to include singletons and not larger open sets that could encompass them
            – IntegrateThis
            Nov 18 at 3:51








          • 1




            @IntegrateThis Note that given a $alpha < omega_1$ the open interval $( alpha , alpha+2)$ is the singleton ${ alpha+1 }$, so these singletons are open. If $mathcal{B}$ is a base, for every such $alpha+1$ there must be some $U in mathcal{B}$ with $alpha+1 in U subseteq { alpha+1 }$, meaning $U = { alpha + 1 }$. (More generally, call an open subset $U$ of a topological space $X$ minimally open if there is no open set $V$ satisfying $emptyset subsetneq V subsetneq U$. Then a base for a topological space must include all of the minimally open subsets.)
            – stochastic randomness
            Nov 18 at 6:08


















          up vote
          2
          down vote













          I think the following qualifies as a counterexample . . .



          Let $k$ be a cardinal number with $k > c$, where $c=|mathbb{R}|$.



          Let $I=[0,1]$ with the usual topology, and let $X$ be the product of $k$ copies of $I$.



          Then





          • $X$ is compact and Hausdorff.$\[4pt]$


          • $X$ is locally compact.$\[4pt]$


          • $X$ is covered by itself, so $X$ qualifies as being covered by a countable union of compact sets.$\[4pt]$

          • The cardinality of the topology on $X$ is greater than $c$ (since $k > c$).$\[4pt]$


          But $X$ is not second countable.



          Suppose instead that $X$ had a countable base, $B$ say.



          Then since every open subset of $X$ is the union of some subset of $B$, it would follow that the cardinality of the topology on $X$ is at most the cardinality of the power set of $B$, hence at most $c$, contradiction.






          share|cite|improve this answer

















          • 1




            Right. And $kgtmathfrak c$ is more than you need: if $k$ is an uncountable cardinal, then $I^k$ is not even first countable. If $kgtmathfrak c$ then $I^k$ is not separable.
            – bof
            Nov 15 at 6:04











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          StackExchange.ready(function() {
          var channelOptions = {
          tags: "".split(" "),
          id: "69"
          };
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function() {
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled) {
          StackExchange.using("snippets", function() {
          createEditor();
          });
          }
          else {
          createEditor();
          }
          });

          function createEditor() {
          StackExchange.prepareEditor({
          heartbeatType: 'answer',
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader: {
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          },
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          });


          }
          });














           

          draft saved


          draft discarded


















          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999201%2fx-is-locally-compact-hausdorff-and-has-a-countable-family-of-compact-subsets%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown

























          2 Answers
          2






          active

          oldest

          votes








          2 Answers
          2






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes








          up vote
          3
          down vote



          accepted










          It is not even true that compact Hausdorff spaces are second-countable. (Of course, a compact Hausdorff space is locally compact and can be covered by a countable family of compact subsets.)



          An example is the ordinal space $[ 0 , omega_1 ]$ (where $omega_1$ denotes the least uncountable ordinal).




          • As a linearly-ordered space, $[0 , omega_1 ]$ is Hausdorff.

          • Since $[0 , omega_1]$ has a greatest element, it is compact (this is essentially because there are no infinite strictly decreasing sequences of ordinals).

          • It is not second-countable because any base must include the singletons ${ alpha }$ where $alpha < omega_1$ is a successor ordinal, and there are uncountably many of these.


          More information about this space can be found on the following post on Dan Ma's Topology Blog:




          • The First Uncountable Ordinal


          (If you want a non-compact example of a locally compact σ-compact Hausdorff space that is not second-countable you can just take a disjoint union of countably-infinite copies of $[0,omega_1]$.)






          share|cite|improve this answer























          • So my conclusion is false?
            – IntegrateThis
            Nov 15 at 5:32






          • 1




            @IntegrateThis Correct: it is not true that every locally compact, σ-compact Hausdorff space is second-countable.
            – stochastic randomness
            Nov 15 at 5:55










          • +1............ $[0,omega_1]$ is not even first-countable.
            – DanielWainfleet
            Nov 15 at 7:52










          • Why dooes the base have to include singletons and not larger open sets that could encompass them
            – IntegrateThis
            Nov 18 at 3:51








          • 1




            @IntegrateThis Note that given a $alpha < omega_1$ the open interval $( alpha , alpha+2)$ is the singleton ${ alpha+1 }$, so these singletons are open. If $mathcal{B}$ is a base, for every such $alpha+1$ there must be some $U in mathcal{B}$ with $alpha+1 in U subseteq { alpha+1 }$, meaning $U = { alpha + 1 }$. (More generally, call an open subset $U$ of a topological space $X$ minimally open if there is no open set $V$ satisfying $emptyset subsetneq V subsetneq U$. Then a base for a topological space must include all of the minimally open subsets.)
            – stochastic randomness
            Nov 18 at 6:08















          up vote
          3
          down vote



          accepted










          It is not even true that compact Hausdorff spaces are second-countable. (Of course, a compact Hausdorff space is locally compact and can be covered by a countable family of compact subsets.)



          An example is the ordinal space $[ 0 , omega_1 ]$ (where $omega_1$ denotes the least uncountable ordinal).




          • As a linearly-ordered space, $[0 , omega_1 ]$ is Hausdorff.

          • Since $[0 , omega_1]$ has a greatest element, it is compact (this is essentially because there are no infinite strictly decreasing sequences of ordinals).

          • It is not second-countable because any base must include the singletons ${ alpha }$ where $alpha < omega_1$ is a successor ordinal, and there are uncountably many of these.


          More information about this space can be found on the following post on Dan Ma's Topology Blog:




          • The First Uncountable Ordinal


          (If you want a non-compact example of a locally compact σ-compact Hausdorff space that is not second-countable you can just take a disjoint union of countably-infinite copies of $[0,omega_1]$.)






          share|cite|improve this answer























          • So my conclusion is false?
            – IntegrateThis
            Nov 15 at 5:32






          • 1




            @IntegrateThis Correct: it is not true that every locally compact, σ-compact Hausdorff space is second-countable.
            – stochastic randomness
            Nov 15 at 5:55










          • +1............ $[0,omega_1]$ is not even first-countable.
            – DanielWainfleet
            Nov 15 at 7:52










          • Why dooes the base have to include singletons and not larger open sets that could encompass them
            – IntegrateThis
            Nov 18 at 3:51








          • 1




            @IntegrateThis Note that given a $alpha < omega_1$ the open interval $( alpha , alpha+2)$ is the singleton ${ alpha+1 }$, so these singletons are open. If $mathcal{B}$ is a base, for every such $alpha+1$ there must be some $U in mathcal{B}$ with $alpha+1 in U subseteq { alpha+1 }$, meaning $U = { alpha + 1 }$. (More generally, call an open subset $U$ of a topological space $X$ minimally open if there is no open set $V$ satisfying $emptyset subsetneq V subsetneq U$. Then a base for a topological space must include all of the minimally open subsets.)
            – stochastic randomness
            Nov 18 at 6:08













          up vote
          3
          down vote



          accepted







          up vote
          3
          down vote



          accepted






          It is not even true that compact Hausdorff spaces are second-countable. (Of course, a compact Hausdorff space is locally compact and can be covered by a countable family of compact subsets.)



          An example is the ordinal space $[ 0 , omega_1 ]$ (where $omega_1$ denotes the least uncountable ordinal).




          • As a linearly-ordered space, $[0 , omega_1 ]$ is Hausdorff.

          • Since $[0 , omega_1]$ has a greatest element, it is compact (this is essentially because there are no infinite strictly decreasing sequences of ordinals).

          • It is not second-countable because any base must include the singletons ${ alpha }$ where $alpha < omega_1$ is a successor ordinal, and there are uncountably many of these.


          More information about this space can be found on the following post on Dan Ma's Topology Blog:




          • The First Uncountable Ordinal


          (If you want a non-compact example of a locally compact σ-compact Hausdorff space that is not second-countable you can just take a disjoint union of countably-infinite copies of $[0,omega_1]$.)






          share|cite|improve this answer














          It is not even true that compact Hausdorff spaces are second-countable. (Of course, a compact Hausdorff space is locally compact and can be covered by a countable family of compact subsets.)



          An example is the ordinal space $[ 0 , omega_1 ]$ (where $omega_1$ denotes the least uncountable ordinal).




          • As a linearly-ordered space, $[0 , omega_1 ]$ is Hausdorff.

          • Since $[0 , omega_1]$ has a greatest element, it is compact (this is essentially because there are no infinite strictly decreasing sequences of ordinals).

          • It is not second-countable because any base must include the singletons ${ alpha }$ where $alpha < omega_1$ is a successor ordinal, and there are uncountably many of these.


          More information about this space can be found on the following post on Dan Ma's Topology Blog:




          • The First Uncountable Ordinal


          (If you want a non-compact example of a locally compact σ-compact Hausdorff space that is not second-countable you can just take a disjoint union of countably-infinite copies of $[0,omega_1]$.)







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited Nov 15 at 6:09

























          answered Nov 15 at 5:20









          stochastic randomness

          1687




          1687












          • So my conclusion is false?
            – IntegrateThis
            Nov 15 at 5:32






          • 1




            @IntegrateThis Correct: it is not true that every locally compact, σ-compact Hausdorff space is second-countable.
            – stochastic randomness
            Nov 15 at 5:55










          • +1............ $[0,omega_1]$ is not even first-countable.
            – DanielWainfleet
            Nov 15 at 7:52










          • Why dooes the base have to include singletons and not larger open sets that could encompass them
            – IntegrateThis
            Nov 18 at 3:51








          • 1




            @IntegrateThis Note that given a $alpha < omega_1$ the open interval $( alpha , alpha+2)$ is the singleton ${ alpha+1 }$, so these singletons are open. If $mathcal{B}$ is a base, for every such $alpha+1$ there must be some $U in mathcal{B}$ with $alpha+1 in U subseteq { alpha+1 }$, meaning $U = { alpha + 1 }$. (More generally, call an open subset $U$ of a topological space $X$ minimally open if there is no open set $V$ satisfying $emptyset subsetneq V subsetneq U$. Then a base for a topological space must include all of the minimally open subsets.)
            – stochastic randomness
            Nov 18 at 6:08


















          • So my conclusion is false?
            – IntegrateThis
            Nov 15 at 5:32






          • 1




            @IntegrateThis Correct: it is not true that every locally compact, σ-compact Hausdorff space is second-countable.
            – stochastic randomness
            Nov 15 at 5:55










          • +1............ $[0,omega_1]$ is not even first-countable.
            – DanielWainfleet
            Nov 15 at 7:52










          • Why dooes the base have to include singletons and not larger open sets that could encompass them
            – IntegrateThis
            Nov 18 at 3:51








          • 1




            @IntegrateThis Note that given a $alpha < omega_1$ the open interval $( alpha , alpha+2)$ is the singleton ${ alpha+1 }$, so these singletons are open. If $mathcal{B}$ is a base, for every such $alpha+1$ there must be some $U in mathcal{B}$ with $alpha+1 in U subseteq { alpha+1 }$, meaning $U = { alpha + 1 }$. (More generally, call an open subset $U$ of a topological space $X$ minimally open if there is no open set $V$ satisfying $emptyset subsetneq V subsetneq U$. Then a base for a topological space must include all of the minimally open subsets.)
            – stochastic randomness
            Nov 18 at 6:08
















          So my conclusion is false?
          – IntegrateThis
          Nov 15 at 5:32




          So my conclusion is false?
          – IntegrateThis
          Nov 15 at 5:32




          1




          1




          @IntegrateThis Correct: it is not true that every locally compact, σ-compact Hausdorff space is second-countable.
          – stochastic randomness
          Nov 15 at 5:55




          @IntegrateThis Correct: it is not true that every locally compact, σ-compact Hausdorff space is second-countable.
          – stochastic randomness
          Nov 15 at 5:55












          +1............ $[0,omega_1]$ is not even first-countable.
          – DanielWainfleet
          Nov 15 at 7:52




          +1............ $[0,omega_1]$ is not even first-countable.
          – DanielWainfleet
          Nov 15 at 7:52












          Why dooes the base have to include singletons and not larger open sets that could encompass them
          – IntegrateThis
          Nov 18 at 3:51






          Why dooes the base have to include singletons and not larger open sets that could encompass them
          – IntegrateThis
          Nov 18 at 3:51






          1




          1




          @IntegrateThis Note that given a $alpha < omega_1$ the open interval $( alpha , alpha+2)$ is the singleton ${ alpha+1 }$, so these singletons are open. If $mathcal{B}$ is a base, for every such $alpha+1$ there must be some $U in mathcal{B}$ with $alpha+1 in U subseteq { alpha+1 }$, meaning $U = { alpha + 1 }$. (More generally, call an open subset $U$ of a topological space $X$ minimally open if there is no open set $V$ satisfying $emptyset subsetneq V subsetneq U$. Then a base for a topological space must include all of the minimally open subsets.)
          – stochastic randomness
          Nov 18 at 6:08




          @IntegrateThis Note that given a $alpha < omega_1$ the open interval $( alpha , alpha+2)$ is the singleton ${ alpha+1 }$, so these singletons are open. If $mathcal{B}$ is a base, for every such $alpha+1$ there must be some $U in mathcal{B}$ with $alpha+1 in U subseteq { alpha+1 }$, meaning $U = { alpha + 1 }$. (More generally, call an open subset $U$ of a topological space $X$ minimally open if there is no open set $V$ satisfying $emptyset subsetneq V subsetneq U$. Then a base for a topological space must include all of the minimally open subsets.)
          – stochastic randomness
          Nov 18 at 6:08










          up vote
          2
          down vote













          I think the following qualifies as a counterexample . . .



          Let $k$ be a cardinal number with $k > c$, where $c=|mathbb{R}|$.



          Let $I=[0,1]$ with the usual topology, and let $X$ be the product of $k$ copies of $I$.



          Then





          • $X$ is compact and Hausdorff.$\[4pt]$


          • $X$ is locally compact.$\[4pt]$


          • $X$ is covered by itself, so $X$ qualifies as being covered by a countable union of compact sets.$\[4pt]$

          • The cardinality of the topology on $X$ is greater than $c$ (since $k > c$).$\[4pt]$


          But $X$ is not second countable.



          Suppose instead that $X$ had a countable base, $B$ say.



          Then since every open subset of $X$ is the union of some subset of $B$, it would follow that the cardinality of the topology on $X$ is at most the cardinality of the power set of $B$, hence at most $c$, contradiction.






          share|cite|improve this answer

















          • 1




            Right. And $kgtmathfrak c$ is more than you need: if $k$ is an uncountable cardinal, then $I^k$ is not even first countable. If $kgtmathfrak c$ then $I^k$ is not separable.
            – bof
            Nov 15 at 6:04















          up vote
          2
          down vote













          I think the following qualifies as a counterexample . . .



          Let $k$ be a cardinal number with $k > c$, where $c=|mathbb{R}|$.



          Let $I=[0,1]$ with the usual topology, and let $X$ be the product of $k$ copies of $I$.



          Then





          • $X$ is compact and Hausdorff.$\[4pt]$


          • $X$ is locally compact.$\[4pt]$


          • $X$ is covered by itself, so $X$ qualifies as being covered by a countable union of compact sets.$\[4pt]$

          • The cardinality of the topology on $X$ is greater than $c$ (since $k > c$).$\[4pt]$


          But $X$ is not second countable.



          Suppose instead that $X$ had a countable base, $B$ say.



          Then since every open subset of $X$ is the union of some subset of $B$, it would follow that the cardinality of the topology on $X$ is at most the cardinality of the power set of $B$, hence at most $c$, contradiction.






          share|cite|improve this answer

















          • 1




            Right. And $kgtmathfrak c$ is more than you need: if $k$ is an uncountable cardinal, then $I^k$ is not even first countable. If $kgtmathfrak c$ then $I^k$ is not separable.
            – bof
            Nov 15 at 6:04













          up vote
          2
          down vote










          up vote
          2
          down vote









          I think the following qualifies as a counterexample . . .



          Let $k$ be a cardinal number with $k > c$, where $c=|mathbb{R}|$.



          Let $I=[0,1]$ with the usual topology, and let $X$ be the product of $k$ copies of $I$.



          Then





          • $X$ is compact and Hausdorff.$\[4pt]$


          • $X$ is locally compact.$\[4pt]$


          • $X$ is covered by itself, so $X$ qualifies as being covered by a countable union of compact sets.$\[4pt]$

          • The cardinality of the topology on $X$ is greater than $c$ (since $k > c$).$\[4pt]$


          But $X$ is not second countable.



          Suppose instead that $X$ had a countable base, $B$ say.



          Then since every open subset of $X$ is the union of some subset of $B$, it would follow that the cardinality of the topology on $X$ is at most the cardinality of the power set of $B$, hence at most $c$, contradiction.






          share|cite|improve this answer












          I think the following qualifies as a counterexample . . .



          Let $k$ be a cardinal number with $k > c$, where $c=|mathbb{R}|$.



          Let $I=[0,1]$ with the usual topology, and let $X$ be the product of $k$ copies of $I$.



          Then





          • $X$ is compact and Hausdorff.$\[4pt]$


          • $X$ is locally compact.$\[4pt]$


          • $X$ is covered by itself, so $X$ qualifies as being covered by a countable union of compact sets.$\[4pt]$

          • The cardinality of the topology on $X$ is greater than $c$ (since $k > c$).$\[4pt]$


          But $X$ is not second countable.



          Suppose instead that $X$ had a countable base, $B$ say.



          Then since every open subset of $X$ is the union of some subset of $B$, it would follow that the cardinality of the topology on $X$ is at most the cardinality of the power set of $B$, hence at most $c$, contradiction.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 15 at 5:38









          quasi

          35.9k22562




          35.9k22562








          • 1




            Right. And $kgtmathfrak c$ is more than you need: if $k$ is an uncountable cardinal, then $I^k$ is not even first countable. If $kgtmathfrak c$ then $I^k$ is not separable.
            – bof
            Nov 15 at 6:04














          • 1




            Right. And $kgtmathfrak c$ is more than you need: if $k$ is an uncountable cardinal, then $I^k$ is not even first countable. If $kgtmathfrak c$ then $I^k$ is not separable.
            – bof
            Nov 15 at 6:04








          1




          1




          Right. And $kgtmathfrak c$ is more than you need: if $k$ is an uncountable cardinal, then $I^k$ is not even first countable. If $kgtmathfrak c$ then $I^k$ is not separable.
          – bof
          Nov 15 at 6:04




          Right. And $kgtmathfrak c$ is more than you need: if $k$ is an uncountable cardinal, then $I^k$ is not even first countable. If $kgtmathfrak c$ then $I^k$ is not separable.
          – bof
          Nov 15 at 6:04


















           

          draft saved


          draft discarded



















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2999201%2fx-is-locally-compact-hausdorff-and-has-a-countable-family-of-compact-subsets%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          Popular posts from this blog

          Plaza Victoria

          Puebla de Zaragoza

          Musa