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- Summary:
- Tychonoff Theorem

Hi,

In Chapter 5 Munkres proves the Tychonoff Theorem and after proving the theorem the first exercise is: Let ##X## be a space. Let ##\mathcal{D}## be a collection of subsets of ##X## that is maximal with respect to finite intersection property

(a) Show that ##x\in\overline{D}## for every ##D\in\mathcal{D}## if and only if every neighborhood of ##x## belongs to ##\mathcal{D}##.

(b) Let ##D\in\mathcal{D}##. Show that if ##A\supset D##, the ##A\in\mathcal{D}##.

(c) Show that if ##X## satisfies the ##T_1## axiom, there is at most one point belonging to ##\bigcap_{D\in\mathcal{D}}\overline{D}##

(a) and (b) are trivial consequences of the lemma 37.2 proven before the Tychonoff Theorem's proof. On the other hand, (c) would be of the same level of triviality if ##T_2## was assumed, however, as written only ##T_1## is assumed.

Could it be just a Typo? or the problem can be solved assuming only ##T_1##. Any comments or help will be appreciated.

In Chapter 5 Munkres proves the Tychonoff Theorem and after proving the theorem the first exercise is: Let ##X## be a space. Let ##\mathcal{D}## be a collection of subsets of ##X## that is maximal with respect to finite intersection property

(a) Show that ##x\in\overline{D}## for every ##D\in\mathcal{D}## if and only if every neighborhood of ##x## belongs to ##\mathcal{D}##.

(b) Let ##D\in\mathcal{D}##. Show that if ##A\supset D##, the ##A\in\mathcal{D}##.

(c) Show that if ##X## satisfies the ##T_1## axiom, there is at most one point belonging to ##\bigcap_{D\in\mathcal{D}}\overline{D}##

(a) and (b) are trivial consequences of the lemma 37.2 proven before the Tychonoff Theorem's proof. On the other hand, (c) would be of the same level of triviality if ##T_2## was assumed, however, as written only ##T_1## is assumed.

Could it be just a Typo? or the problem can be solved assuming only ##T_1##. Any comments or help will be appreciated.

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