Editing Topological vector space (section)

===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>&nbsp;of}}&nbsp;increasing&nbsp;nonempty&nbsp;[[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>&nbsp;of}}&nbsp;increasing&nbsp;nonempty&nbsp;[[Chain (order theory)|chain]]
|-
!style="text-align:left;"|Arbitrary&nbsp;unions (of&nbsp;at&nbsp;least&nbsp;1&nbsp;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&nbsp;unions (of&nbsp;at&nbsp;least&nbsp;1&nbsp;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>&nbsp;of}}&nbsp;decreasing&nbsp;nonempty&nbsp;[[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>&nbsp;of}}&nbsp;decreasing&nbsp;nonempty&nbsp;[[Chain (order theory)|chain]]
|-
!style="text-align:left;"|Arbitrary&nbsp;intersections (of&nbsp;at&nbsp;least&nbsp;1&nbsp;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&nbsp;intersections (of&nbsp;at&nbsp;least&nbsp;1&nbsp;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
|}