/sci/ - Science & Math

This wasn't exactly hard, imo, but I think it was a cool problem. It was in my topology course's exam:

Let <span class="math">X[/spoiler] be a connected space of atleast two points, satisfying separation axioms <span class="math">T_1[/spoiler] and <span class="math">T_4[/spoiler]. Show that <span class="math">X[/spoiler] is uncountable.

Because the space satisfies <span class="math">T_1[/spoiler], every singleton set is closed in it (this was proved to be equal to the axiom in the material). Because the space has atleast two points, there exist such <span class="math"> x, y \in X[/spoiler] that <span class="math">x \neq y[/spoiler]. Their singletons are closed and disjoint, and the space is <span class="math">T_4[/spoiler], so it satisfies the conditions of Urysohn's lemma, giving us a continuous function mapping x to 0 and y to 1.

Because the space is connected and the function continuous, the image of the space, with respect to the function, is connected. As the only connected sets on the real line are the empty set, singletons, intervals and the set itself, it must be one of them. It cannot be the empty set, because <span class="math">X[/spoiler] is not empty, nor can it be a singleton because of the function. The function is from <span class="math">X[/spoiler] to the unit interval, so the image can't be the real line either. It must be an interval, and it must contain 0 and 1. Thus the image of our space is the unit interval itself.

Because the function is onto, the cardinality of <span class="math">X[/spoiler] is greater or equal to the cardinality of the unit interval. As the unit interval is uncountable, so too must be <span class="math">X[/spoiler].

>>7238861

>>7123709
>>7123709

>>6735699

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