Show there exists a minimal topology that makes a given function continuous

Let $f:X \rightarrow Y$ be a function where $Y$ is a topological space. Show that there exists a topology $\tau$ in $X$ such that:

1. It makes $f$ continuous.

2. If $\tau'$ is a different topology in $X$ that makes $f$ continuous, then every open set in $\tau$ is open in $\tau'$.

I think that $\tau=\{X, \emptyset\};$

1. Let $\tau_2$ be the topology defined in $Y$. Then $f$ is continuous iff for every $O \in \tau_2$, $f^{-1}(O)\in\tau$. $f^{-1}(O)=\{x\in X : f(x) \in O\}.$ Now since $\tau_2$ is a topology, then $O$ is either $\emptyset$ of a subset of Y. So if $O=\emptyset$ then $f^{-1}\{\emptyset\} = \emptyset\in\tau$. What I can't prove is the other case. I don't see why if $O$ is a subset of $Y$ then $f^{-1}\{O\}\subset X$ is the whole $X$.

2. It's obvious since $\tau=\{X, \emptyset\}$ is the less fine topology in $X$

Edit: As it was pointed out my choice of $\tau$ was wrong. Now let's consider $\tau=\{f^{-1}(U):U\text{ open in }Y\}.$ First, let's see it is a topology:

• $f^{-1}(\emptyset)=\emptyset$ and $f^{-1}(Y)=X$, so $\emptyset, X \in\tau.$

• Let $\{O_i\}_{i \in I}$ be a family of open sets in $\tau.$ Then, for each $O_i, O_i = f^{-1}(U_i), U_i\in \tau_2$, so $\bigcup_{i \in I} O_i=\bigcup_{i \in I} f^{-1}(U_i) = f^{-1}(\bigcup_{i \in I} U_i)$ and since $\tau_2$ is a topology, $\bigcup_{i \in I} U_i$ is open in $\tau_2,$ so $\{O_i\}_{i \in I}=f^{-1}(\bigcup_{i \in I} U_i)$ is open in $\tau$.

• Let $O_1,O_2 \in \tau.$ Then $O_1=f^{-1}(U_1), O_2=f^{-1}(U_2), U_1, U_2 \in \tau_2.$ So $O_1\cap O_2=f^{-1}(U_1)\cap f^{-1}(U_2)=f^{-1}(U_1\cap U_2),$ and since $\tau_2$ is a topology and $U_1,U_2$ are open in $\tau_2,$ then $U_1\cap U_2$ is open in $\tau_2,$ so $O_1\cap O_2=f^{-1}(U_1\cap U_2)$ is open in $\tau.$

Now:

1. It's clear that given $f:X \rightarrow Y,$ $\tau=\{f^{-1}(U):U\text{ open in }Y\}$ makes $f$ continuous since, by definition, $f:(X, \tau_1) \rightarrow (Y, \tau_2)$ is continuous if and only if for each $U$ open in $\taụ_2, f^{-1}(U)$ is open in $\tau_1.$

2. Let $\tau'$ be a topology on $X$. Then $\tau'$ will make $f:(X, \tau') \rightarrow (Y, \tau_2)$ continuous if and only if for every $U$ open in $\tau_2,$ $f^{-1}(U)$ is open in $\tau_1.$ This means that $\tau'$ has to contain, at least, those subsets $O_i \subset X$ such that $O_i=f^{-1}(U_i), U_i \in Y,$ and the family of these subsets, $\{O_i\}_{i \in I}$ is precisely $\tau,$ so $\tau \subset \tau'$ for every $\tau'$ that makes any $f$ continuous.

• Your choice for the topology is wrong: take for example $f:\{0,1\}\rightarrow\{0,1\}$ to be the identity, then if in the topology for $Y$ the singleton subsets are open their inverse images will no longer be open in $X$ with the topology you described above. You should have a topology on $X$ which contains at least the inverse images of all open subsets of $Y$ (can you guess how to describe it?) – TheMadcapLaughs Nov 30 '17 at 23:00
• @HennoBrandsma edited. – Yagger Dec 1 '17 at 13:05

The topology $\{\emptyset,X\}$ on $X$ is often called the indiscrete topology.

The function $f$ need not be continuous if you endow $X$ with the indiscrete topology. Consider, for instance, $X=\{0,1\}$ and $Y$ the reals with the usual topology.

Then the map $f\colon X\to Y$ defined by $f(0)=0$ and $f(1)=1$ is not continuous with respect to the indiscrete topology on $X$, because $f^{-1}\bigl((0,\infty)\bigr)=\{1\}$ is not open.

So your conjecture is doomed, unfortunately.

Since $f$ must be continuous with respect to $\tau$, you see that $f^{-1}(U)$ should belong to $\tau$ for every open set $U$ in $Y$. Now prove that $$\tau=\{f^{-1}(U):U\text{ open in }Y\}$$ is indeed the required topology on $X$

• what if we take the discrete topology on $X$, where everey subset of $X$ is open? – palio Dec 1 '17 at 8:04
• @palio The discrete topology need not satisfy the second requirement. Consider the identity over a set with more than two elements; endow the codomain with the indiscrete topology. – egreg Dec 1 '17 at 9:12
• Thanks! I updated the post proving the topology you provided is indeed the one that verifies the two statements. – Yagger Dec 1 '17 at 12:16

The topology is $\tau=\{f^{-1}(G): G~\text{open in }Y\}$, and it is the smallest one which makes $f:X\rightarrow Y$ being continuous.

Topology you are considering is not correct.

So correct Topology is given by,

$\tau=${$f^{-1}(A):A\in \tau_2$}, where $\tau_2$ is the topology on $Y$.

1) By the definition of continuous function, $f$ is continuous.

2) Again by the definition of continuous function and construction of topology, $\tau$ is weaker than any other topology wrt which $f$ is continuous.