Difference between revisions of "Discrete metric and topology/Summary"

Latest revision as of 13:04, 27 July 2015

Property	Comment
induced topology	discrete topology - which is the topology $(X,\mathcal{P}(X))$ (where $\mathcal{P}$ denotes power set)
Open ball	$B_r(x):=\{p\in X\vert\ d(p,x)< r\}=\left\{\begin{array}{lr}\{x\} & \text{if }r\le 1 \\ X & \text{otherwise}\end{array}\right.$
Open sets	Every subset of $X$ is open. Proof outline: as for a subset $A\subseteq X$ we can show $\forall x\in A\exists r[B_r(x)\subseteq A]$ by choosing say, that is $A$ contains an open ball centred at each point in $A$ .
Connected	The topology generated by $(X,d_\text{discrete})$ is not connected if $X$ has more than one point. Proof outline: Let $A$ be any non empty subset of $X$ , then define $B:=A^c$ which is also a subset of $X$ , thus $B$ is open. Then $A\cap B=\emptyset$ and $A\cup B=X$ thus we have found a separation, a partition of non-empty disjoint open sets, that separate the space. Thus it is not connected if $X$ has only one point then we cannot have a partition of non empty disjoint sets. Thus it cannot be not connected, it is connected.

Revision as of 13:03, 27 July 2015 (view source) Alec (Talk \| contribs) m (Alec moved page Discrete and metric topology/Summary to Creating Discrete metric and topology/Summary without leaving a redirect: Wrong title) ← Older edit	Latest revision as of 13:04, 27 July 2015 (view source) Alec (Talk \| contribs) m (Alec moved page Creating Discrete metric and topology/Summary to Discrete metric and topology/Summary without leaving a redirect: Failed to copy and paste)
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