Псевдотопологическое пространство

Псевдотопологи́ческое простра́нство — множество с дополнительной предельной структурой определённого типа (так называемой псевдотопологией). Исторически понятие псевдотопологического пространства появилось как обобщение топологического пространства. Псевдотопологические пространства были введены в 1959 г. Фишером ^[1]. Псевдотопологические пространства естественным образом возникают при построении дифференциального исчисления в пространствах без нормы.^[2] Топологические пространства можно рассматривать как частные случаи псевдотопологических.^[3]

Определение

Псевдотопологическое пространство $E$ представляет собой множество $M$ , наделённое псевдотопологией. Множество $M$ называется несущим множеством пространства $E$ и обозначается через ${\underline {E}}$ . Если фильтр $\mathrm {X}$ в $E$ сходится к точке $x$ в данной псевдотопологии, то это обозначается как $\mathrm {X} \downarrow _{x}E$ . Псевдотопология в $M$ определяется заданием для каждого $x\in M$ некоторого семейства фильтров в $M$ , удовлетворяющих следующим условиям:

Если фильтр сходится к $x$ , то к $x$ сходится и любой меньший фильтр. $\mathrm {X} \downarrow _{x}E,\Psi \leqslant \mathrm {X} \Rightarrow \Psi \downarrow _{x}E$
Если два фильтра сходятся к $x$ , то к $x$ сходится и их верхняя грань. $\mathrm {X} _{1}\downarrow _{x}E,\mathrm {X} _{2}\downarrow _{x}E\Rightarrow \mathrm {X} _{1}\vee \mathrm {X} _{2}\downarrow _{x}E$
Фильтр $\left[x\right]$ сходится к $x$ . $\left[x\right]\downarrow _{x}E$

Примечания

↑ Fisher H. R., Limesraume, Math. Ann., 137 (1959), 269-303
↑ Фрёлихер, 1970, с. 6.
↑ Фрёлихер, 1970, с. 21.

Литература

Фрёлихер, А., Бухер В. Дифференциальное исчисление в векторных пространствах без нормы. — М.: Мир, 1970.

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