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Let $\{(X_{n},\tau_{n})\}_{n\in \mathbb{N}}$ be a sequence of topological spaces, such that for every $n \in \mathbb{N}$, $(X_{n}, \tau_{n})$ is a subspace of $(X_{n+1}, \tau_{n+1})$ for all $n \in \mathbb{N}$. Let $X = \bigcup_{n \in \mathbb{N}} X_{n}$.

Let $\tau = \lbrace A \subseteq X | (\forall n \in \mathbb{N}) A \cap X_{n} \in \tau_{n} \rbrace$.

  1. Prove that $(X, \tau)$ is a topological space.

  2. Prove that $(X_{n}, \tau_{n})$ is a subspace of $(X, \tau)$ for every $n \in \mathbb{N}$.

  3. Prove that a function $f:X \to Y$ is continuous ($Y$ is some topological space) iff $f|_{X_n}$ is continuous for all $n \in \mathbb{N}$.

  4. Prove that $(X, \tau)$ is $T_{1}$ iff $(\forall n \in \mathbb{N}) (X_{n}, \tau_{n})$ is $T_{1}$.

I've proven 1. and 3. pretty easily, the first by definition, and the third by noticing that for any $B \in \tau_{Y}$, $f^{-1}(B) = \bigcup_{n \in \mathbb{N}} f^{-1}|_{X_{n}}(B)$.

However, I'm having trouble with 2. and 4. For 2, I need to prove the following equivalence: $A \in \tau_{n} \iff \exists B \in \tau : A = B \cap X_{n}$. I'm having trouble with $\implies$: I've tried picking $B_{m} \in \tau_{m}$ for $m \geq n$ such that $A=B_{m} \cap X_{n}$, and then setting $B = \bigcup_{m \geq n} B_{m}$, but I can't prove that $B \cap X_{j} \in \tau_{j}$ for all $j$ (if $j>n$, then I can't prove that $B \cap X_{j} = \bigcup_{m \in \mathbb{N}} (B_{m} \cap X_{j})$ is open in $\tau_{j}$, because for $m < j$, I don't know if $B_{m}$ are open in $\tau_{j}$).

As far as 4, I haven't gotten any further than applying the definition directly: if $x, y \in X$, $x \neq y$, then $x, y \in X_{m}$ starting with $m= m_{0}$. So then if $X_{m}$ are $T_{1}$, I get $U_{m}$, $V_{m}$ such that $x \in U_{m} \setminus V_{m}$, $y \in V_{m} \setminus U_{m}$. So I'd like to have $\bigcup_{m \geq m_{0}} U_{m}$ and $\bigcup_{m \geq m_{0}} U_{m}$ be open in $\tau$, but I don't know how to prove that either.

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  • $\begingroup$ For 3) suppose each $X_n$ is $T_1$ and let $x \neq y$ in X. There exists an integer N such that x and y both belong to $X_N$. ( The fact that $X_n \subset X_{n+1}$ is used here). Let U be an open set in $X_N$ containing x but not y. Then U is also open in X and it contains x but not y. Similarly there exists an open set V in X containing y but not x. Converse part: suppose X is $T_1$. Fix N and let $x \neq y$ in $X_N$. Let U be an open set in X containing x but not y. Then $U \cap X_N$ is open in $X_N$, it contains x but not y. Similarly there is an open set in $X_N$ containing y but not x. $\endgroup$ – Kavi Rama Murthy Jan 10 '18 at 5:53
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For part 2, choose the $B_i $ nested, where $B_n=B_{n+1}\cap X_n $. This choice is possible since each $X_n $ is a subspace of $X_{n+1} $. It then follows that $B\cap X_j=B_j$.

For part 4, recall that $T_1$ is equivalent to points being closed, since clearly if points are closed then $V=X\setminus \{x\}$ and $U=X\setminus \{y\}$ are the desired open sets, and for $X $ $T_1$ we have for each $y\in X $, $y\neq x $, an open $U (y) $ not containing $x $, and hence $\bigcup_{y\neq x}U (y)=X\setminus \{x\} $ is open.

Now, let $U=X\setminus \{x\}$. Then $U\cap X_n= X_n\setminus \{x\}$ which is open by assumption, so $U $ is open and therefore $\{x\}$ is closed.

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  • $\begingroup$ If $X$ is $T_1$ all its subspaces are, so all $X_n$ are. Just as a remark for the other direction of 4. $\endgroup$ – Henno Brandsma Jan 10 '18 at 5:27
  • $\begingroup$ @HennoBrandsma thanks, I missed the second f in iff. $\endgroup$ – Noah Riggenbach Jan 10 '18 at 5:28

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