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For any subsets <math>G \subseteq T</math> and <math>N \subseteq Y,</math> let
<math display="block">\mathcal{U}(G, N) := \{f \in F : f(G) \subseteq N\}.</math>
'''Properties'''▼
Assume that <math>G \in \mathcal{G}</math> and <math>N \in \mathcal{N}.</math>▼
<math>\mathcal{U}(G, N)</math> is an [[Absorbing set|absorbing]] subset of <math>F</math> if and only if for all <math>f \in F,</math> <math>N</math> absorbs <math>f(G)</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}} ▼
If <math>N</math> is [[Balanced set|balanced]]{{sfn|Narici|Beckenstein|2011|pp=371-423}} (respectively, [[Convex set|convex]]) then so is <math>\mathcal{U}(G, N).</math>▼
<ul>▼
<li>If <math>s</math> is a scalar then <math>s \mathcal{U}(G, N) = \mathcal{U}(G, s N),</math> so that in particular, <math>- \mathcal{U}(G, N) = \mathcal{U}(G, - N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>▼
<li><math>\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N)</math> for any subsets <math>G, H \subseteq X</math> and any non-empty subsets <math>M, N \subseteq Y.</math>{{sfn|Jarchow|1981|pp=43-55}} For such subsets, it follows that,▼
<ul>▼
<li>if <math>M \subseteq N</math> then <math>\mathcal{U}(G, M) \subseteq \mathcal{U}(G, N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>▼
<li>if <math>G \subseteq H</math> then <math>\mathcal{U}(H, N) \subseteq \mathcal{U}(G, N).</math></li>▼
<li>For any <math>M, N \in \mathcal{N}</math> and subsets <math>G, H, K</math> of <math>T,</math> if <math>G \cup H \subseteq K</math> then <math>\mathcal{U}(K, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N).</math></li>▼
</ul>▼
</li>▼
<li><math>\mathcal{U}(\varnothing, N) = F.</math></li>▼
<li><math>\mathcal{U}(G, N) - \mathcal{U}(G, N) \subseteq \mathcal{U}(G, N - N).</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}</li>▼
<li><math>\mathcal{U}(G, M) + \mathcal{U}(G, N) \subseteq \mathcal{U}(G, M + N).</math>{{sfn|Jarchow|1981|pp=43-55}}</li>▼
<li>For any family <math>\mathcal{S}</math> of subsets of <math>T</math> and any family <math>\mathcal{M}</math> of neighborhoods of the origin in <math>Y,</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}} <math display="block">\mathcal{U}\left(\bigcup_{S \in \mathcal{S}} S, N\right) = \bigcap_{S \in \mathcal{S}} \mathcal{U}(S, N) \qquad \text{ and } \qquad \mathcal{U}\left(G, \bigcap_{M \in \mathcal{M}} M\right) = \bigcap_{M \in \mathcal{M}} \mathcal{U}(G, M).</math></li>▼
</ul>▼
===𝒢-topology===
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that is, if and only if for every <math>G \in \mathcal{G}</math> and every <math>f \in F,</math> <math>f(G)</math> is [[Bounded set (topological vector space)|bounded]] in <math>Y.</math>
}}
▲'''Properties'''
▲
▲Then <math>\mathcal{U}(G, N)</math> is an [[Absorbing set|absorbing]] subset of <math>F</math> if and only if for all <math>f \in F,</math> <math>N</math> absorbs <math>f(G)</math>.{{sfn|Narici|Beckenstein|2011|pp=371-423}}
▲If <math>N</math> is [[Balanced set|balanced]]{{sfn|Narici|Beckenstein|2011|pp=371-423}} (respectively, [[Convex set|convex]]) then so is <math>\mathcal{U}(G, N).</math>
▲<ul>
▲<li>If <math>s</math> is a scalar then <math>s \mathcal{U}(G, N) = \mathcal{U}(G, s N),</math> so that in particular, <math>- \mathcal{U}(G, N) = \mathcal{U}(G, - N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
▲<li><math>\mathcal{U}(G \cup H, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N)</math> for any subsets <math>G, H \subseteq X</math> and any non-empty subsets <math>M, N \subseteq Y.</math>{{sfn|Jarchow|1981|pp=43-55}} For such subsets, it follows that,
▲<ul>
▲<li>if <math>M \subseteq N</math> then <math>\mathcal{U}(G, M) \subseteq \mathcal{U}(G, N).</math>{{sfn|Narici|Beckenstein|2011|pp=371-423}}</li>
▲<li>if <math>G \subseteq H</math> then <math>\mathcal{U}(H, N) \subseteq \mathcal{U}(G, N).</math></li>
▲<li>For any <math>M, N \in \mathcal{N}</math> and subsets <math>G, H, K</math> of <math>T,</math> if <math>G \cup H \subseteq K</math> then <math>\mathcal{U}(K, M \cap N) \subseteq \mathcal{U}(G, M) \cap \mathcal{U}(H, N).</math></li>
▲</ul>
▲</li>
▲<li><math>\mathcal{U}(\varnothing, N) = F.</math></li>
▲<li><math>\mathcal{U}(G, N) - \mathcal{U}(G, N) \subseteq \mathcal{U}(G, N - N).</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}}</li>
▲<li><math>\mathcal{U}(G, M) + \mathcal{U}(G, N) \subseteq \mathcal{U}(G, M + N).</math>{{sfn|Jarchow|1981|pp=43-55}}</li>
▲<li>For any family <math>\mathcal{S}</math> of subsets of <math>T</math> and any family <math>\mathcal{M}</math> of neighborhoods of the origin in <math>Y,</math>{{sfn|Narici|Beckenstein|2011|pp=19-45}} <math display="block">\mathcal{U}\left(\bigcup_{S \in \mathcal{S}} S, N\right) = \bigcap_{S \in \mathcal{S}} \mathcal{U}(S, N) \qquad \text{ and } \qquad \mathcal{U}\left(G, \bigcap_{M \in \mathcal{M}} M\right) = \bigcap_{M \in \mathcal{M}} \mathcal{U}(G, M).</math></li>
▲</ul>
===Uniform structure===
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