Why in uniform space contains small sets should be nonempty meets?

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In the figure, $delta(F)<V$ means $Ftimes Fsubseteq V$. Wy $bigcapmathcal Fneqvarnothing$?
If we let $mathcal F=x,y$, then it clearly satisfies the condition: contains arbitrary small sets, but $bigcapmathcal F=varnothing$ .enter image description here




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    "contains at most one point" means "contain zero point" (ie "is empty") OR "contains one point". So your example doesn't contradict the statement.
    – Suzet
    Jul 19 at 5:37











  • Is the projection operator $p_i$ from a product of uniform spaces to $X_i$ is a closed map? i.e., if $FsubseteqPi X_i$ is closed, then $p_i(F)$ is closed in $X_i$ induced by $mathcal U_i$.
    – Shen Chong
    Jul 20 at 4:46














up vote
0
down vote

favorite












In the figure, $delta(F)<V$ means $Ftimes Fsubseteq V$. Wy $bigcapmathcal Fneqvarnothing$?
If we let $mathcal F=x,y$, then it clearly satisfies the condition: contains arbitrary small sets, but $bigcapmathcal F=varnothing$ .enter image description here




general-topology




share|cite|improve this question















  • 1




    "contains at most one point" means "contain zero point" (ie "is empty") OR "contains one point". So your example doesn't contradict the statement.
    – Suzet
    Jul 19 at 5:37











  • Is the projection operator $p_i$ from a product of uniform spaces to $X_i$ is a closed map? i.e., if $FsubseteqPi X_i$ is closed, then $p_i(F)$ is closed in $X_i$ induced by $mathcal U_i$.
    – Shen Chong
    Jul 20 at 4:46












up vote
0
down vote

favorite









up vote
0
down vote

favorite











In the figure, $delta(F)<V$ means $Ftimes Fsubseteq V$. Wy $bigcapmathcal Fneqvarnothing$?
If we let $mathcal F=x,y$, then it clearly satisfies the condition: contains arbitrary small sets, but $bigcapmathcal F=varnothing$ .enter image description here




general-topology




share|cite|improve this question











In the figure, $delta(F)<V$ means $Ftimes Fsubseteq V$. Wy $bigcapmathcal Fneqvarnothing$?
If we let $mathcal F=x,y$, then it clearly satisfies the condition: contains arbitrary small sets, but $bigcapmathcal F=varnothing$ .enter image description here





general-topology





share|cite|improve this question










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asked Jul 19 at 5:21









Shen Chong

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  • 1




    "contains at most one point" means "contain zero point" (ie "is empty") OR "contains one point". So your example doesn't contradict the statement.
    – Suzet
    Jul 19 at 5:37











  • Is the projection operator $p_i$ from a product of uniform spaces to $X_i$ is a closed map? i.e., if $FsubseteqPi X_i$ is closed, then $p_i(F)$ is closed in $X_i$ induced by $mathcal U_i$.
    – Shen Chong
    Jul 20 at 4:46












  • 1




    "contains at most one point" means "contain zero point" (ie "is empty") OR "contains one point". So your example doesn't contradict the statement.
    – Suzet
    Jul 19 at 5:37











  • Is the projection operator $p_i$ from a product of uniform spaces to $X_i$ is a closed map? i.e., if $FsubseteqPi X_i$ is closed, then $p_i(F)$ is closed in $X_i$ induced by $mathcal U_i$.
    – Shen Chong
    Jul 20 at 4:46







1




1




"contains at most one point" means "contain zero point" (ie "is empty") OR "contains one point". So your example doesn't contradict the statement.
– Suzet
Jul 19 at 5:37





"contains at most one point" means "contain zero point" (ie "is empty") OR "contains one point". So your example doesn't contradict the statement.
– Suzet
Jul 19 at 5:37













Is the projection operator $p_i$ from a product of uniform spaces to $X_i$ is a closed map? i.e., if $FsubseteqPi X_i$ is closed, then $p_i(F)$ is closed in $X_i$ induced by $mathcal U_i$.
– Shen Chong
Jul 20 at 4:46




Is the projection operator $p_i$ from a product of uniform spaces to $X_i$ is a closed map? i.e., if $FsubseteqPi X_i$ is closed, then $p_i(F)$ is closed in $X_i$ induced by $mathcal U_i$.
– Shen Chong
Jul 20 at 4:46















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