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"Exercise 1. Show that if $X$ is equipped with the discrete metric $d$ then every subset of $X$ is both open and closed. Deduce that any function $f : (X, d) → (Y, dY )$ is continuous."

My lecturer shared the following answer:

"Exercise 1. If $A\subset{X}$ then for every $a\in{A}$ we have $B(a, 1/2) = \{a\}$, and so $A$ is open. Since every subset is open, every subset is also closed. The function $f$ is continuous if $f^{-1}(U)$ is open in $(X,d)$ for any open set U in $(Y,dY )$; but $f^{-1}(U)$ is a subset of $(X,d)$, so is always open."

So my question is that are singletons open? I thought they are closed. Or does it depend on the metric?

kam
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    It depends on the topology, though that can be induced by the metric if you have one. Generally in a topology you declare all the sets that you consider open, so you can have some, all, or none of the singletons open – postmortes May 13 '19 at 09:46
  • Ok thank you. When you say induced by the metric, are referring to the topology? – kam May 13 '19 at 09:47
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    Yes: a metric gives you a way to define open sets, and so you can generate ("induce") a topology from it – postmortes May 13 '19 at 09:49
  • Singletons are not necessarily closed in general topological spaces, though... – YuiTo Cheng May 13 '19 at 09:50
  • So just to clarify. You can give a metric, then from this we can check which subsets are open and hence induce the topology? Does a metric always induce a topology? – kam May 13 '19 at 09:51
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    Yes, a metric always induces a topology, but not every topological space is metrizable. – YuiTo Cheng May 13 '19 at 09:52
  • By metrisable, I assume you mean able to define a metric on the space in question? – kam May 13 '19 at 09:55
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    It is common to teach metric spaces without mentioning "topology", defining everything (such as open and closed sets, continuity, etc.) directly in terms of the metric. This generally works well. Once you get to general topological spaces it then turns out that almost everything can be defined through the topology, i.e. just by knowing which sets are open. – hmakholm left over Monica May 13 '19 at 09:57
  • Note that openness and closedness are not necessarily mutually exclusive, though they can be provided the set in question is a nontrivial, proper subset of a connected space. – Kaj Hansen May 13 '19 at 09:58
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    "Metrizable" is used about a topology and means that this particular topology can be induced by a metric. – hmakholm left over Monica May 13 '19 at 09:58
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    One point of topology is that it generalises features which were found useful in metric spaces to a wider range of spaces - amongst other things it allows for discussion of continuity in contexts where there is no notion of distance. – Mark Bennet May 13 '19 at 10:23

1 Answers1

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In the case of discrete metric all sets are both open and closed. In particular singletons are open and closed. In general singletons in a metric space are closed. They need not be open as in the case of the real line with usual metric.

Kavi Rama Murthy
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