IDs 200-299

Any bijective function is a homeomorphism in discrete space. Just take the permutation that swaps $a$ and $b$.

Essentially the same proof as Fréchet Urysohn $\implies$ Sequential (T184). Just swap “sequential closure” with “radial closure”.

Fréchet Urysohn is stronger because it asserts there is a sequence, a.k.a a transfinite sequence of length $\omega$.

Any radially closed set must be, in particular, sequentially closed.

Any open set has more than one point.

Suppose a set $A$ is sequentially closed. Let $p \in X$ and $(x_\alpha)_{\alpha < \lambda} \subseteq A$. Take $V$ a countable nbd of $p$. If $x_\alpha \to p$, then there is an ordinal $\mu$ for which $x_\alpha \in V$ for all $\alpha \ge \mu$. The set of ordinals $\mu \le \alpha < \lambda$ is isomorphic to some $\lambda'$. So we have $(y_\alpha)_{\alpha < \lambda'} \subseteq V$, still with $y_\alpha \to V$.

From a converging transfinite sequence within a countable set, we can extract a $\omega$-sequence that still converges to the same value, this was done in (T211). So we construct a sequence $(z_n)$ with $z_n \to p$, and so $p \in A$ as we wished to prove.

Given $p \in \overline{A}$, choose $D \subseteq A$ countable with $p \in \overline{D}$. Suppose $(x_\alpha)_{\alpha < \lambda}$ is some transfinite sequence in $D$ with $x_\alpha \to p$. We now construct a sequence in $D$ which converges to $p$ and we’re done (thanks to PatrickR).

If some $y \in D$ is cofinal in $(x_\alpha)$, this means any neighborhood $U$ of $x$ must contain $y$, so we can construct a constant sequence $y_n = y$ with $y_n \to p$. If no $y \in D$ is cofinal, then each $I_y = \{ \alpha < \lambda \ : \ x_\alpha = y \}$ is finite and $\lambda = \bigcup_{y \in D} I_y$ is a countable union of finite sets, so it is a countable ordinal. We can assume $\lambda$ to be a regular cardinal, so it is already a sequence.

If each local basis $\mathcal{V}_x$ is countable and $X$ is countable, then $\mathcal{B} = \bigcup_{x \in X} \mathcal{V}_x$ is a countable basis.

$\{x\}$ is a finite neighborhood of $x$.

In particular, singletons are discrete.

Compact sets are countably compact.

Globally implies locally.

This is just “Has a countable $k$-network $\implies$ Has a $\sigma$-locally finite $k$-network” (T352) with $T_3$ added.

$\mathcal{P}(X) = \{\emptyset, X\}$ if and only if it has 0 or 1 point.

There’s only one possible topology.

There’s only one possible topology.

If you have an infinite amount of apples, then you have at least 2 apples.

Define $d(x, y) = 1$ if $x,y$ belong to the same set in the partition, and 0 otherwise. This is a pseudometric.

Some point has a neighborhood not containing another point.

A space is a subspace of itself.

Singletons form a network.

Every metric is a pseudometric.

By contraposition, if $d(x, y) = 0$, then they both have the same neighborhoods.

Globally implies locally.

A countable basis is a local basis for every point.

$T_2$ is $T_0$ with $R_1$.

Any two distinct points are distinguishable in a $T_0$ space.

Immediate by definition.

By definition, $T_1$ is $T_0$ and $R_0$.

$T_0$ ensures any two points are distinguishable. This is an if and only if.

Important result to solve the Riemann hypothesis.

It has finitely many self-maps (continuous or not).