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===Properties preserved by set operators=== * The balanced hull of a compact (respectively, [[totally bounded]], open) set has that same property.{{sfn|Narici|Beckenstein|2011|pp=67-113}} * The [[Minkowski sum|(Minkowski) sum]] of two compact (respectively, bounded, balanced, convex) sets has that same property.{{sfn|Narici|Beckenstein|2011|pp=67-113}} But the sum of two closed sets need {{em|not}} be closed. * The convex hull of a balanced (resp. open) set is balanced (respectively, open). However, the convex hull of a closed set need {{em|not}} be closed.{{sfn|Narici|Beckenstein|2011|pp=67-113}} And the convex hull of a bounded set need {{em|not}} be bounded. The following table, the color of each cell indicates whether or not a given property of subsets of <math>X</math> (indicated by the column name, "convex" for instance) is preserved under the set operator (indicated by the row's name, "closure" for instance). If in every TVS, a property is preserved under the indicated set operator then that cell will be colored green; otherwise, it will be colored red. So for instance, since the union of two absorbing sets is again absorbing, the cell in row "<math>R \cup S</math>" and column "Absorbing" is colored green. But since the arbitrary intersection of absorbing sets need not be absorbing, the cell in row "Arbitrary intersections (of at least 1 set)" and column "Absorbing" is colored red. If a cell is not colored then that information has yet to be filled in. {| class="wikitable mw-collapsible mw-collapsed" |+ Properties preserved by set operators !rowspan="2"|Operation !colspan="100"|Property of <math>R,</math> <math>S,</math> and any other subsets of <math>X</math> that is considered |- ![[Absorbing set|Absorbing]] ![[Balanced set|Balanced]] ![[Convex set|Convex]] ![[Symmetric set|Symmetric]] !Convex<br />Balanced !Vector<br />subspace !Open !Neighborhood<br />of 0 !Closed !Closed<br />Balanced !Closed<br />Convex !Closed<br />Convex<br />Balanced ![[Barrelled set|Barrel]] !Closed<br />Vector<br />subspace ![[Totally bounded|Totally<br />bounded]] ![[Compact set|Compact]] !Compact<br />Convex ![[Relatively compact]] ![[Complete space|Complete]] ![[Sequentially complete space|Sequentially<br />Complete]] ![[Banach disk|Banach<br />disk]] ![[Bounded set (topological vector space)|Bounded]] ![[Bornivorous set|Bornivorous]] ![[Infrabornivorous]] ![[Nowhere dense set|Nowhere<br />dense]] (in <math>X</math>) ![[Meagre set|Meager]] ![[Separable space|Separable]] ![[Metrizable TVS|Pseudometrizable]] !Operation |- !style="text-align:left;"|<math>R \cup S</math> |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{na}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{na}}<!--Convex Balanced--> |{{na}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |{{na}}<!--Closed Convex Balanced--> |style="background:;"|<!--Barrel--> |{{na}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{na}}<!--Compact convex--> |{{ya}}<!--Relatively compact--> |style="background:;"|<!--Complete--> |{{ya}}<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{ya}}<!--Meager--> |{{ya}}<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|<math>R \cup S</math> |- !style="text-align:left;"|{{nowrap|<math>\cup</math> of}} increasing nonempty [[Chain (order theory)|chain]] |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |{{na}}<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |{{na}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{na}}<!--Closed Vector subspace--> |{{na}}<!--Totally bounded--> |{{na}}<!--Compact--> |{{na}}<!--Compact convex--> |{{na}}<!--Relatively compact--> |{{na}}<!--Complete--> |{{na}}<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{na}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|{{nowrap|<math>\cup</math> of}} increasing nonempty [[Chain (order theory)|chain]] |- !style="text-align:left;"|Arbitrary unions (of at least 1 set) |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{na}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{na}}<!--Convex Balanced--> |{{na}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |{{na}}<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |{{na}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{na}}<!--Closed Vector subspace--> |{{na}}<!--Totally bounded--> |{{na}}<!--Compact--> |{{na}}<!--Compact convex--> |{{na}}<!--Relatively compact--> |{{na}}<!--Complete--> |{{na}}<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{na}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|Arbitrary unions (of at least 1 set) |- !style="text-align:left;"|<math>R \cap S</math> |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |{{ya}}<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{ya}}<!--Meager--> |style="background:;"|<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|<math>R \cap S</math> |- !style="text-align:left;"|{{nowrap|<math>\cap</math> of}} decreasing nonempty [[Chain (order theory)|chain]] |{{na}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Open--> |{{na}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |style="background:;"|<!--Compact--> |style="background:;"|<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |style="background:;"|<!--Meager--> |style="background:;"|<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|{{nowrap|<math>\cap</math> of}} decreasing nonempty [[Chain (order theory)|chain]] |- !style="text-align:left;"|Arbitrary intersections (of at least 1 set) |{{na}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{na}}<!--Open--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |style="background:;"|<!--Compact--> |style="background:;"|<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |style="background:;"|<!--Meager--> |style="background:;"|<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|Arbitrary intersections (of at least 1 set) |- !style="text-align:left;"|<math>R + S</math> |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |style="background:;"|<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |style="background:;"|<!--Closed Convex Balanced--> |style="background:;"|<!--Barrel--> |style="background:;"|<!--Closed Vector subspace--> |style="background:;"|<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |style="background:;"|<!--Nowhere dense--> |style="background:;"|<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|<math>R + S</math> |- !style="text-align:left;"|Scalar multiple |{{na}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Open--> |{{na}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |{{na}}<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |{{na}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{na}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |{{ya}}<!--Relatively compact--> |{{ya}}<!--Complete--> |{{ya}}<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |{{na}}<!--Bornivorous--> |{{na}}<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{ya}}<!--Meager--> |{{ya}}<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|Scalar multiple |- !style="text-align:left;"|Non-0 scalar multiple |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |{{ya}}<!--Relatively compact--> |{{ya}}<!--Complete--> |{{ya}}<!--Sequentially complete--> |{{ya}}<!--Banach disk--> |{{ya}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{ya}}<!--Meager--> |{{ya}}<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|Non-0 scalar multiple |- !style="text-align:left;"|Positive scalar multiple |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |{{ya}}<!--Complete--> |{{ya}}<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{ya}}<!--Meager--> |{{ya}}<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|Positive scalar multiple |- !style="text-align:left;"|[[Closure (topology)|Closure]] |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |{{ya}}<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |style="background:;"|<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|[[Closure (topology)|Closure]] |- !style="text-align:left;"|[[Interior (topology)|Interior]] |{{na}}<!--Absorbing--> |{{na}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |style="background:;"|<!--Convex Balanced--> |{{na}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |{{na}}<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |{{na}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{na}}<!--Closed Vector subspace--> |style="background:;"|<!--Totally bounded--> |{{na}}<!--Compact--> |{{na}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|[[Interior (topology)|Interior]] |- !style="text-align:left;"|[[Balanced core]] |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |style="background:;"|<!--Compact--> |style="background:;"|<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{ya}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|[[Balanced core]] |- !style="text-align:left;"|[[Balanced hull]] |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{na}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |style="background:;"|<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{na}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |{{ya}}<!--Banach disk--> |{{ya}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|[[Balanced hull]] |- !style="text-align:left;"|[[Convex hull]] |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |style="background:;"|<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |style="background:;"|<!--Totally bounded--> |style="background:;"|<!--Compact--> |{{ya}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |{{ya}}<!--Banach disk--> |{{na}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|[[Convex hull]] |- !style="text-align:left;"|[[Convex balanced hull]] |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |style="background:;"|<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |style="background:;"|<!--Totally bounded--> |style="background:;"|<!--Compact--> |style="background:;"|<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |{{ya}}<!--Banach disk--> |{{na}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|[[Convex balanced hull]] |- !style="text-align:left;"|Closed balanced hull |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{na}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |style="background:;"|<!--Totally bounded--> |style="background:;"|<!--Compact--> |style="background:;"|<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|Closed balanced hull |- !style="text-align:left;"|Closed convex hull |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |style="background:;"|<!--Totally bounded--> |style="background:;"|<!--Compact--> |style="background:;"|<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{na}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|Closed convex hull |- !style="text-align:left;"|Closed convex balanced hull |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |style="background:;"|<!--Totally bounded--> |style="background:;"|<!--Compact--> |style="background:;"|<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{na}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|Closed convex balanced hull |- !style="text-align:left;"|[[Linear span]] |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |style="background:;"|<!--Closed--> |style="background:;"|<!--Closed Balanced--> |style="background:;"|<!--Closed Convex--> |style="background:;"|<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{na}}<!--Totally bounded--> |{{na}}<!--Compact--> |{{na}}<!--Compact convex--> |{{na}}<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |{{ya}}<!--Banach disk--> |{{na}}<!--Bounded--> |{{ya}}<!--Bornivorous--> |{{ya}}<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |{{na}}<!--Meager--> |style="background:;"|<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|[[Linear span]] |- !style="text-align:left;"|Pre-image under a continuous linear map |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{ya}}<!--Open--> |{{ya}}<!--Neighborhood of 0--> |{{ya}}<!--Closed--> |{{ya}}<!--Closed Balanced--> |{{ya}}<!--Closed Convex--> |{{ya}}<!--Closed Convex Balanced--> |{{ya}}<!--Barrel--> |{{ya}}<!--Closed Vector subspace--> |{{na}}<!--Totally bounded--> |{{na}}<!--Compact--> |{{na}}<!--Compact convex--> |{{na}}<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{na}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |style="background:;"|<!--Nowhere dense--> |style="background:;"|<!--Meager--> |{{na}}<!--Separable--> |{{na}}<!--Pseudometrizable--> !style="text-align:left;"|Pre-image under a continuous linear map |- !style="text-align:left;"|Image under a continuous linear map |{{na}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |{{na}}<!--Open--> |{{na}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |{{na}}<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |{{na}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{na}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |style="background:;"|<!--Meager--> |{{ya}}<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|Image under a continuous linear map |- !style="text-align:left;"|Image under a continuous linear surjection |{{ya}}<!--Absorbing--> |{{ya}}<!--Balanced--> |{{ya}}<!--Convex--> |{{ya}}<!--Symmetric--> |{{ya}}<!--Convex Balanced--> |{{ya}}<!--Vector subspace--> |style="background:;"|<!--Open--> |style="background:;"|<!--Neighborhood of 0--> |style="background:;"|<!--Closed--> |style="background:;"|<!--Closed Balanced--> |style="background:;"|<!--Closed Convex--> |style="background:;"|<!--Closed Convex Balanced--> |style="background:;"|<!--Barrel--> |style="background:;"|<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{ya}}<!--Compact--> |{{ya}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |style="background:;"|<!--Complete--> |style="background:;"|<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |style="background:;"|<!--Bornivorous--> |style="background:;"|<!--Infrabornivorous--> |{{na}}<!--Nowhere dense--> |style="background:;"|<!--Meager--> |{{ya}}<!--Separable--> |style="background:;"|<!--Pseudometrizable--> !style="text-align:left;"|Image under a continuous linear surjection |- !style="text-align:left;"|Non-empty subset of <math>R</math> |{{na}}<!--Absorbing--> |{{na}}<!--Balanced--> |{{na}}<!--Convex--> |{{na}}<!--Symmetric--> |{{na}}<!--Convex Balanced--> |{{na}}<!--Vector subspace--> |{{na}}<!--Open--> |{{na}}<!--Neighborhood of 0--> |{{na}}<!--Closed--> |{{na}}<!--Closed Balanced--> |{{na}}<!--Closed Convex--> |{{na}}<!--Closed Convex Balanced--> |{{na}}<!--Barrel--> |{{na}}<!--Closed Vector subspace--> |{{ya}}<!--Totally bounded--> |{{na}}<!--Compact--> |{{na}}<!--Compact convex--> |style="background:;"|<!--Relatively compact--> |{{na}}<!--Complete--> |{{na}}<!--Sequentially complete--> |style="background:;"|<!--Banach disk--> |{{ya}}<!--Bounded--> |{{na}}<!--Bornivorous--> |{{na}}<!--Infrabornivorous--> |{{ya}}<!--Nowhere dense--> |{{ya}}<!--Meager--> |style="background:;"|<!--Separable--> |{{ya}}<!--Pseudometrizable--> !style="text-align:left;"|Non-empty subset of <math>R</math> |- !Operation ![[Absorbing set|Absorbing]] ![[Balanced set|Balanced]] ![[Convex set|Convex]] ![[Symmetric set|Symmetric]] !Convex<br />Balanced !Vector<br />subspace !Open !Neighborhood<br />of 0 !Closed !Closed<br />Balanced !Closed<br />Convex !Closed<br />Convex<br />Balanced ![[Barrelled set|Barrel]] !Closed<br />Vector<br />subspace ![[Totally bounded|Totally<br />bounded]] ![[Compact set|Compact]] !Compact<br />Convex ![[Relatively compact]] ![[Complete space|Complete]] ![[Sequentially complete space|Sequentially<br />Complete]] ![[Banach disk|Banach<br />disk]] ![[Bounded set (topological vector space)|Bounded]] ![[Bornivorous set|Bornivorous]] ![[Infrabornivorous]] ![[Nowhere dense set|Nowhere<br />dense]] (in <math>X</math>) ![[Meagre set|Meager]] ![[Separable space|Separable]] ![[Metrizable TVS|Pseudometrizable]] !Operation |}
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