Let $\tau$ be the sets $U \subset \Bbb R$ for which $U = \emptyset$ or $\Bbb R \setminus U$ is countable. Show that $\tau$ is a topology for $\Bbb R$.
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Show "countable complement topology" is a topology
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How to prove that a finite set is countable
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Are these bases for a topology?
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Is the cofinite topology on an uncountable set first countable?
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How to prove that $\mathbb{Q}$ ( the rationals) is a countable set
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Finite Cartesian Product of Countable sets is countable?
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Showing that a topological space is ${\rm T}_1$
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Help with a proof. Countable sets.
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Is the cofinite topology on a countable set first countable?
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Real Analysis - Prove a set is countable
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Is $[0,1]$ a countable disjoint union of closed sets?
12472
Let $X$ and $Y$ be countable sets. Then $X\cup Y$ is countable
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$\mathbb{R}$ with the lower limit topology is not second-countable
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What is the cardinality of the set of all topologies on $\mathbb{R}$?
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How can I prove that compact subset iff sequentially compact in second countable topological spaces?
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Example 2, Sec. 24, in Munkres' TOPOLOGY, 2nd ed: If $X$ is a well ordered set, then $X \times [0, 1)$ is a linear continuum
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Let $(X,\tau)$ be a connected topological space, $Y\subset X$ connected and $U\subset X\setminus Y$ clopen. Prove that $ Y \cup U$ is connected