Articles of product space

Continuity of product of fuctions w.r.t. product and box topology

Let $f:A\rightarrow\prod_{\alpha\in J} X_\alpha$ be given by the equation $f(a)=(f_\alpha (a))_{\alpha \in J}$ where $f_{\alpha}:A\rightarrow X_\alpha$ for each $\alpha$. Let $\Pi X_\alpha$ have the box topology. Show that the implication; “the function $f$ is continuous if each $f_\alpha$ is continuous” is not true for this topology. How do I prove this? Can anyone help? Obviously […]

Is projection of a closed set $F\subseteq X\times Y$ always closed?

If we have closed subset $F$ of product $X \times Y$ (product topology) does it mean that $p_1(F)$ (projection on first coordinate) is closed in $X$ and $p_2(F)$ in $Y$ are closed? If not, why not (some counterexample or explanation please =)

Showing that the product and metric topology on $\mathbb{R}^n$ are equivalent

I’m new to topology, and can’t figure out why the metric and product topologies over $\mathbb{R}^n$ are equivalent. Could someone please show me how to prove this?

$X$ is homeomorphic to $X\times X$ (TIFR GS $2014$)

Question is : Suppose $X$ is a topological space of infinite cardinality which is homeomorphic to $X\times X$. Then which of the following is true: $X$ is not connected. $X$ is not compact $X$ is not homeomorphic to a subset of $\mathbb{R}$ None of the above. I guess first two options are false. We do […]

Uncountable product in the category of metric spaces.

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesn’t possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where $X\in\mathrm{Ob(Met)}$, $\pi_\nu\in \mathrm{Hom_{Met}}(X,X_\nu)$ is called a product of family of metric spaces $\{X_\nu:\nu\in\Lambda\}$ if for each $Y\in\mathrm{Ob(Met)}$ and $\{\varphi_\nu:\nu\in\Lambda\}$ where $\varphi_\nu\in \mathrm{Hom_{Met}}(Y,X_\nu)$ there exist unique $\varphi\in\mathrm{Hom_{Met}}(Y,X)$ such that $\varphi_\nu=\pi_\nu\varphi$. […]

Compact topological space not having Countable Basis?

Does there exist a compact topological space not having countable basis? I have constructed a product space from uncountably many unit intervals $[0,1]$, endowed with the product topology. Tychonoff’s Theorem shows that this topological space is compact Hausdorff space, but I’m not sure how to prove that this space does not have any countable basis. […]

Product of totally disconnected space is totally disconnected?

I read that the cartesian product with the product topology of a family of totally disconnected topological spaces is totally disconnected, too. Is that true? How are the connected components in the cartesian product with the product topology defined? Didn’t find it yet. Thanks.

In general, why is the product topology not equal to the box topology

I am trying to understand a counter-example showing that the box topology and product topology are not equal. Here it is: Let $\tau$ and $\tau’$ be the product and box topologies respectively. Let $X_i = \mathbb{R}\ \forall i$ and let $U_i = (-1,1)\ \forall i$. Then $U:= \prod_{i=1}^{\infty}U_i$ and $U \in \tau’$ but $U \notin […]

The product of Hausdorff spaces is Hausdorff

I’m confused how it can be true that the product of an infinite number of Hausdorff spaces $X_\alpha$ can be Hausdorff. If $\prod_{\alpha \in J} X_\alpha$ is a product space with product topology, the basis elements consists of of products $\prod_{\alpha \in J} U_{\alpha}$ where $U_{\alpha}$ would equal $X_{\alpha}$ for all but finitely many $\alpha$’s. […]

Separability of a product metric space

I am trying to prove the following: ‘If $(X_1,d_1)$ and $(X_2,d_2)$ are separable metric spaces (that is, they have a countable dense subset), then the product metric space $X_1 \times X_2$ is separable.’ It seems pretty straightforward, but I would really appreciate it if someone could verify that my proof works. Since $(X_1,d_1)$ and $(X_2,d_2)$ […]