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Suppose $X, Y $ are topological spaces. Pick a point $(x,y) \in X \times Y$. I am trying to show that $\{x\} \times Y $ is homeomorphic to $Y$. Is this true?

Is this function $f(x,y) = y$ a homeomorphism? This is obviously continuous since it is projection map. But its inverse is also continuous? thanks for your help

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    $\begingroup$ Its inverse is continuous if and only if the two components $y \mapsto x$ and $y \mapsto y$ are continuous. $\endgroup$ – Daniel Fischer Oct 1 '13 at 9:12
  • $\begingroup$ Of course it is since $\{x\}$ is just a meaningless symbol associated to $Y$. $\endgroup$ – Shuchang Oct 1 '13 at 11:44
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Here are some methods to show that the map $g:Y\to\{x\}\times Y,\ y\mapsto(x,y)$, which is the inverse of $f$, is continuous. The second one is especially useful.

  • You can deal directly with the open sets. An non-empty open set in $\{x\}\times Y$ has the form $\{x\}\times U$ for all open sets $U$ in $Y.$ What is its preimage under $g$ ?
  • A map from a space $Z$ to a product $\prod_{i\in I} Y_i$ is continuous iff each component is continuous. In your case $g$ has two components, one is the constant map $Y\to\{x\}$, the other is the identity on $Y$.
  • If you happen to know a bit category theory, there is an elegant way: A singleton space $\{x\}$ is a terminal object in the category Top, and the product space is the categorical product. Now, in each category, the canonical map from an object to its product with the terminal object is a natural isomorphism.
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