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- Homework Statement
- Prove that a locally constant function is constant on a connected topological space X.

- Relevant Equations
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Let $X$ be a topological space and ##Y## a set. A function ##f: X \to Y## is said to be locally constant if, for every ##x \in X##, there is an open set ##U## containing ##x## so that the restriction ##f|_U: U \to Y## is constant. Prove that if ##X## is connected, a locally constant function on ##X## is constant.

Proof: ##X## is a connected topological space and ##f:X\rightarrow Y## is a locally constant function from ##X## to a set ##Y##. A function is locally constant iff ##\forall x_0\in X,## there exists a neighborhood ##U## of ##x## so that ##\forall x\in U, f(x)=f(x_0)##. Suppose there exists nonempty open subsets ##U## and ##V## in ##X## so that for all ##x\in U## and ##y\in V## the function restricted to ##U## and ##V## take ##x## and ##y## into different elements in ##Y##, so that ##f|_U(u)=u'## and ##f|_V(v)=v'## and ##u'\neq v##. If ##U\cap V\neq \emptyset,## then the function is not well defined over ##U\cap V## which leads to a contradiction to the assumption that ##f## is defined ##\forall x\in X##. If ##U\cap V= \emptyset##, it implies that ##X## is disconnected, which leads me to a contradiction. ##\therefore f## is constant for any ##x\in X##.

Is my proof correct, and if no, how could I improve it?

Proof: ##X## is a connected topological space and ##f:X\rightarrow Y## is a locally constant function from ##X## to a set ##Y##. A function is locally constant iff ##\forall x_0\in X,## there exists a neighborhood ##U## of ##x## so that ##\forall x\in U, f(x)=f(x_0)##. Suppose there exists nonempty open subsets ##U## and ##V## in ##X## so that for all ##x\in U## and ##y\in V## the function restricted to ##U## and ##V## take ##x## and ##y## into different elements in ##Y##, so that ##f|_U(u)=u'## and ##f|_V(v)=v'## and ##u'\neq v##. If ##U\cap V\neq \emptyset,## then the function is not well defined over ##U\cap V## which leads to a contradiction to the assumption that ##f## is defined ##\forall x\in X##. If ##U\cap V= \emptyset##, it implies that ##X## is disconnected, which leads me to a contradiction. ##\therefore f## is constant for any ##x\in X##.

Is my proof correct, and if no, how could I improve it?